The veteran midfielder spent last season on loan at Goodison Park from Manchester City, and is now a free agent after his contract with the Premier League champions expired.
But despite this, Everton are obliged to pay the Sky Blues £2m to bring in Barry on a full basis due to a clause inserted in the deal to take him to Merseyside last year.
The Toffees are far from keen to do this, while City are unwilling to back down.
This has left the door open for West Ham, who are able to enter discussions with the 33-year-old and bring him in for nothing.
Sam Allardyce has been tasked with playing more attractive football at Upton Park next term, and is thought to see Barry as a good addition his midfield.
The former England international is known to favour Everton, but with the standoff showing little sign of ending, he may be forced to look elsewhere.
His former club Aston Villa are also in the hunt.