Everton and Villa both realise that they need to add to their squads in the summer and are said to feel that Agosto could be a good option.
The 27-year-old goalkeeper has impressed in La Liga this term, but Deportivo are struggling to tie him down to a long-term deal, with his contract expiring next summer.
As a result, the Spanish club acknowledge they may be powerless to stop him leaving when the transfer window re-opens and are determined not to lose him on a free.
If Agosto did move to Everton or Villa it would be his first taste of English football and he would likely have a gentle introduction.
Villa currently have Brad Guzan as their number one, while Shay Given is an able deputy, and Agosto would only be brought in to add competition.
Everton, meanwhile, have the likes of Tim Howard and Joel Robles available to them and Agosto would likely come in as third choice.