I've no idea whether this Daily Express article is authoritative or speculative - it would be helpful to see a similar message in other parts of the mainstream media, not to mention direct quotes from both Mrs May for the UK and Sr Rajoy for Spain in particular, although the article does contain a number of direct statements attributed to a spokesman for Mrs May:
Comments directly attributed to a spokesman for Mrs Theresa May, British Prime Minister:
"They both agreed it was an area it would be good to get agreement early on in the negotiations.
"He said we need to get an agreement on reciprocal rights.
"We are firmly of the opinion that we want this issue that is resolved early. There is some broad agreement across member states but not all of them.
"They both agreed it would be an area it would be good to get an early agreement on."
This sounds hopeful but I'd be cautious about over-optimism until formal agreement is reached - both the UK and Spain have an interest in resolving this issue quickly, but if Germany and some others remain intransigent, it may not happen so smoothly as this implies - my view is and has always been that the rights of existing EU27 residents in the UK must be protected; it is not the British government that is holding cross-border EU residents to ransom, despite attempts in some parts of the British media to paint this picture, it is countries like Germany.
There should be no illusions about this - Brussels regards Britons in the rest of the EU and residents in the UK from the rest of the EU as bargaining chips in their plans to make the UK's exit (aka 'escape') from the clutches of the EU as painful as possible - the Eurozone is in such a mess that Brussels will go to almost any lengths to keep its banks afloat (notably Deutsche and certain of the French banks, which are heavily exposed in places like Spain, Italy and of course Greece, etc).
Personally I have few qualms about Merkel's 'crazy' immigration policy, because luckily the UK is not a member of the Schengen area, something which is only possible because the UK happens to be comprised of islands off the mainland of Europe (Ireland is also outside of the the Schengen area, it being a practical solution because of its long-standing relations with the UK, but is really for Ireland's convenience, especially given the open border between Ireland and Northern Ireland dating back to, from memory, 1923 - long before either country joined the EEC (now EU).
It is no accident that thousands of prospective immigrants surged through Schengen in to Germany, Finland, amongst other destinations, and to Calais (hoping to smuggle themselves into the UK, until the French government at last took the situation in hand and dispersed them around the country). It is possible that a proportion of the large number of 'immigrants' referred to are genuine 'asylum seekers' to whom refuge must be granted, but it seems to be generally agreed that these represent at most 50% of the total numbers and perhaps considerably less. Although the rules for claiming 'asylum' are that a claim must be lodged in the first 'country of sanctuary' (often Greece or Italy within the EU, but also countries such as Turkey and Lebanon), it has always seemed to me only fair that all EU members should share the burden, certainly financially if not always physically - it is striking that the UK contributes more than the rest of the EU combined to this effort and is second only to the US in this financial assistance. Those 'first sanctuary' countries, all of them, have in my view tried their best to accommodate as many genuine refugees as they can, given their own high levels of unemployment and the fact that their economies are generally much less robust than some of the countries further north in the EU (Austria, Germany, Holland, Finland and of course the UK) and from what I can gather the two of them in the EU (Greece and Italy) had been pleading for help from the rest of the EU, both financial and practical, for some years, but had largely been ignored and left to get on with it as best they could, until the surge of numbers in the summer of 2015 shamed the EU and in particular Germany to accept as many as could travel there - the open Schengen border then (since restricted somewhat in practical terms) made transit across the continent fairly easy, if chaotic - but the drawback was that the huge numbers meant that any real effort to distinguish between 'immigrants' ('economic' or other) and genuine 'asylum seekers' became almost impossible, not aided by the fact that many discarded whatever ID they may have had, or obtained forged documentation from what were considered to be more acceptable 'asylum' countries, such as Syria or Iraq and a few others such as Sudan or South Sudan etc, when many of those concerned probably came from completely different countries, whilch although poor were not in a state of political turmoil requiring 'asylum'. This article is NOT about immigration or asylum, however, but it is impossible not to mention this topic when discussing wider matters relating to EU membership and the rights and responsibilities that status implies.
The UK will always need "immigrants" and generally welcomes them, as we have a robust economy and relatively low unemployment. We also have a long history of accepting refugees in need of asylum and I hope this will continue long into the future, quite apart from our international obligations in this regard. It seems to me that although relatively few genuine asylum seekers arrive direct in the UK from the countries they are fleeing from, even if through equally dangerous countries, that we have to play our part in helping, at the very least financially, those countries where genuine asylum seekers are likely arrive first in Europe. It is certainly unfair to leave it all to Italy and Greece within the EU, or Turkey, Jordan and Lebanon, which already have accommodated large numbers of refugees - and are not wealthy countries themselves - simply because geography places them in the path of these genuine refugees.
Coming back to the main topic of this article, however, I do hope the governments of the EU27 overall (and not just Spain, obviously one of the more important member states) will come to their senses and agree quickly measures to protect both EU27 citizens in the UK and UK citizens in the EU27 - the UK government and now apparently Spain want to get this matter resolved quickly. Let's hope sense will prevail in Brussels (& Germany) so that this agreement can de done quickly. Although I didn't agree with 'remainer' Theresa May's seeming intransigence, I have come to the conclusion her robust, but fair, attitude has been the better course to follow and will in due course stand the best chance of allowing this matter to be resolved in a common sense way. But it needs reciprocal good will from the EU27 too - given that, I do not think the UK will be found wanting. But we have taken the decision to depart the EU and the stamping of feet in frustration and anger by anyone is highly unlikely to change this and given the British nature is only likely to harden attitudes here, which frankly is the last thing I wish to see happen.
(NB/ This article is also cross-posted to my "Spanish" blog casabill - the blog [link] - see also link to relevant article).