Galatasaray have been heavily linked with the experienced midfielder in recent days, and it was claimed that the club were considering moving for him before their transfer market closes on February 9.
However, according to ESPN, Eriksen will be staying with Man United for the rest of the 2023-24 campaign, with his future not being decided until the summer.
The report claims that Galatasaray had shown interest but have now accepted that a deal would be very difficult to do at this stage of the market.
Eriksen impressed for Erik ten Hag's side during the 2022-23 campaign after arriving on a free transfer from Brentford.
The 31-year-old has struggled to make his mark this term, though, only managing one goal and two assists in 19 appearances in all competitions.
Eriksen has played 15 times in the Premier League this season, with nine of those being starts, but the return to fitness of Casemiro and emergence of Kobbie Mainoo has pushed him down the pecking order.
The Dane has been an unused substitute for the team's last two Premier League games against Wolverhampton Wanderers and West Ham United, while he also did not feature against Newport County in the FA Cup at the end of last month.
Galatasaray are allegedly preparing to revisit their interest this summer, with Man United expected to be open to offers for the experienced midfielder.
Since arriving at Old Trafford in the summer of 2022, Eriksen has made 63 appearances for the Red Devils, scoring three times and registering 12 assists in the process.
Man United are expected to revamp their midfield this summer, with Casemiro potentially moving on amid interest from the Saudi Pro League.
The club, as it stands, are also unlikely to sign Sofyan Amrabat on a permanent basis from Fiorentina, with the Morocco international finding it difficult to show his best form this season.
A decision will need to be made on Scott McTominay, who came close to leaving last summer but has since become an important player for Ten Hag, scoring seven times in all competitions this term.
Bayern Munich's Joshua Kimmich and Nice's Khephren Thuram are believed to be among the club's midfield targets for this summer's transfer window.