The midfielder, 22, was one of the standout stars of the Premier League last season, scoring eight times in 39 appearances under Patrick Vieira’s tutelage.
His fine form even saw him earn his first caps in Gareth Southgate’s England side, as well as being named Palace’s Player of the Season.
Gallagher had been expected to fight for his place at Stamford Bridge this term after receiving assurances from boss Thomas Tuchel over the summer.
But the Cobham academy graduate started on the bench for their Premier League opener against Everton and was only introduced in stoppage time.
With the World Cup in Qatar fast approaching, the Eagles hope an offer of regular first-team minutes will lure him back to South London for a second loan stint, according to the Mail.
Chelsea are also hoping to sign Frenkie de Jong from Barcelona — a move that would add to Gallagher’s midfield competition, which already includes the likes of Jorginho, Mateo Kovacic and N’Golo Kante.
Sky Sports say the Blues will also move for the Dutchman’s team-mate Pierre-Emerick Aubameyang should the Blaugrana agree to sell both players this month to ease their financial issues.
Meanwhile, 90min report that Newcastle and West Ham are among other clubs to have made enquiries over Gallagher’s availability.
Reports in Spain say Juventus are close to striking a deal with Barca to sign forward Memphis Depay.
Manchester City have agreed to sign left-back Sergio Gomez from Anderlecht, according to Fabrizio Romano.
AC Milan have opened talks with Arsenal over the signing of midfielder Albert Sambi Lokonga, as claimed by reports in Italy.