The Dane returned to football following recovering from a heart condition and signed a short-term deal with Brentford in January.
The 30-year-old has impressed in a Brentford shirt, helping the club to all but secure survival in the Premier League, and the Bees are keen to sign him on a longer-term contract in the summer.
Newcastle want Eriksen and it has been suggested they are confident of a deal, but they face competition from within the Premier League.
It has been claimed that more than three Premier League club are keen to sign Eriksen on a free transfer in the summer.
The Dane will have a number of offers to choose from once the season is finished, with him expected to stay in England next season.
Tottenham have also been linked with a move for him, but it is unclear whether the north London club want him back in the summer.
Eriksen has registered three assists and has scored once for Brentford in nine appearances since joining the club, winning praise for the impact he has made.