On Monday the Toffees announced they had agreed an £8.6million fee with the Dutch club and the 23-year-old travelled to Merseyside to discuss personal terms and undergo the usual routine health checks.
However, Press Association Sport understands during those assessments medical staff identified an issue which raised enough concerns for Everton to go back to Twente in an attempt to renegotiate the deal.
Instead of the full fee up front the Premier League side wanted to pay a smaller amount with the remainder dependent on appearances and other factors.
Twente have rejected the new terms and although negotiations are continuing the move is in the balance. If it is not wrapped up in the next 24 hours Everton will not allow it to drag on into the final day of the transfer window.
Fer missed almost two months of the Eredivisie season after sustaining a knee injury on Holland duty in September.
Everton, who have refused to comment on the situation, are obviously concerned about the long-term prognosis and want to protect themselves against future problems, especially considering the substantial initial fee they were originally prepared to pay.
Twente chairman Joop Munsterman is less than impressed though.
"Everton have additional requirements on the method of payment. Here we do not agree. It's stuck," Munsterman was quoted as saying by De Telegraaf.
"We don't play that game at Twente. They can come up with any demand they wish but we're not playing along with their demands."