It will be the first time the joint sovereignty proposal for Gibraltar, which was raised informally with Mr Cook's predecessor Malcolm Rifkind in January, is put formally at bilateral ministerial talks about the Rock.
"It is a new proposal, a revolutionary process that marks a substantial change in Spain's approach," a Foreign Ministry spokeswoman said yesterday. But she added that Spain's proposal was open-ended and contained no timeframe. If the principle were acceptable, the timing would be for Britain to decide.
"We have never raised the question of leaseback, along the lines of Hong Kong. And we have never talked in terms of a timetable. All we can say is we're talking long term. Our position is that Britain can keep Gibraltar as long as it wants, under the terms of the Treaty of Utrecht."
The Foreign Office dismissed the proposal as unacceptable when it was first trailed in January, saying that it did not take into account the wishes of the Gibraltarians, who are vehement in their desire to remain British.
Perhaps anticipating a similar response today, Mr Matutes is shoring up his proposal with two supporting arguments. First, he proposes to leave open the question of whether or not the proposed joint-sovereignty period would be a transition to eventual recovery of Spanish sovereignty, or whether it could last indefinitely. "We aren't specifying whether it is a transition period or not," the spokeswoman said.
Secondly, the proposal comes with a package of economic and social measures designed to protect the rights of Gibraltarians and improve their material conditions. "Our claim is the territorial integration of Gibraltar into Spain, but we do not want to exercise rights over the Gibraltarian population: we want to annex the territory, not the people," the spokeswoman said. Accordingly, Gibraltarians would be free to choose to stay British or take Spanish nationality.
Details of proposed economic and social incentives remain to be developed, the ministry says, but could amount to generous devolution of powers under a tailor-made statute of autonomy comparable to deals Madrid has fashioned with the Basques and the Catalans.
Spain's plan amounts to a pragmatic, softly-softly assault upon sovereignty, chipping away at areas of political competence with British consent, leaving the eventual outcome open. The aim is to break the stalemate over sovereignty that has dogged regular talks about the Rock between Britain and Spain, of which today's is the latest round: Britain has refused even to discuss sovereignty in this forum.
"Spain is ready to maintain and protect the rights of the Gibraltarians and to talk about a statute of autonomy. Our idea is that during a long period of co-sovereignty, Gibraltarians would make new gains as well as keeping the advantages they already have, and would gradually get used to coexistence with Spain," Mr Matutes said recently, and added: "We respect Britain's sovereignty over the Rock, but we hope to recoup it eventually."
Spain insists it will talk only to Britain, and not to what it calls Gibraltar's "authorities". Mr Matutes has expressed his "esteem" for Mr Cook during "frank and difficult" haggles over Gibraltar in talks over Nato restructuring. Britain last week lifted its reserve on Spain's full integration into Nato's military structures, but still wants Spanish restrictions on access to the Rock lifted. The Nato talks seem to have put new heart in the stalled bilateral talks, whose stalemate Madrid blames mostly on the Gibraltarian Chief Minister, Peter Caruana.
Mr Caruana agrees that the logic of European integration dictates a "modernisation" of Gibraltar's status. "We want to change from being a colony or a dependent territory. We would like to become a crown dependency, like Guernsey and the Isle of Man, which still enables us to keep British sovereignty. We don't want to break our links with the Crown."