Jese is currently plying his trade at Real Madrid, but his lack of first-team experience this term means that Everton and West Ham seem to believe that the Spanish giants may be open to loaning him out.
Jese has struggled to displace the likes of Gareth Bale, Karim Benzema and Cristiano Ronaldo, and Madrid may feel that his development could be better served by getting competitive action.
It means that a move to the Premier League is a possibility and if Everton or West Ham could capture the front man it would be a signal of their intent.
Both West Ham and Everton will hold ambitions of claiming a European place and bringing a player of Jese's quality in would be quite a coup.
West Ham will feel that they can offer Jese more first-team action than Everton, but the Toffees hold the advantage of being able to highlight previous loan deals from Spain which have worked out well.
Roberto Martinez's style encourages technicality and they can point to the progress that Gerard Deulofeu made at Goodison Park to try and convince Jese to join.