The Championship side are keen on a loan deal for the Costa Rican.
The defender is down the pecking order at Goodison Park, with Leighton Baines the first choice left-back by some distance and young defender Brendan Galloway having impressed in the role this season.
Oviedo is also still struggling to maintain his long-term fitness, having picked up a series of smaller injuries after battling back from breaking his leg in January 2014.
He has mostly been used in the cup games by manager Roberto Martinez this season to build up his fitness, but was also used at right back in the 3-3 draw with Chelsea, before being stretchered off with another injury.
Much is likely to rest on Everton’s move for Leeds United youngster Sam Byram, with the defender thought to be close to a move.
He is likely to provide cover at right-back for Seamus Coleman, and could also cover at left-back, which would likely make Roberto Martinez more open to allowing Oviedo to leave.