The Spaniard has been mesmerising out on the wing for the Toffees this term.
The Hungarian stopper is struggling to get first-team football in Spain.
Leicester City signed Daniel Amartey from FC Copenhagen earlier this week.
Bolton Wanderers forward Zach Clough has been linked with a host of clubs including Everton, Swansea City, Sheffield Wednesday and Wolverhampton Wanderers.
Jay Rodriguez has missed much of Southampton's season so far with injury.
Queens Park Rangers have already signed Conor Washington in the January transfer window.
The Belgian has come in for criticism since moving to Merseyside.
Pickford has been given a couple of first team opportunities recently by Sam Allardyce.
Allen scored the penalty that sent Liverpool through to the final of the League Cup but says the side are looking for consistency.
The English market is proving cruel, should Leeds return to Italy?