The Argentina captain is eager to secure a move in January to maintain his match fitness ahead of next year's World Cup.
The MLS season ends in December.
Fotomac is reporting Galatasaray are aware of Messi's plans and monitoring his situation closely.
Gala remain confident about persuading Messi to join on a four-month loan for the second half of this season.
Messi is on £17.6 million a year at Inter Miami, with Gala needing to fund part of that wage package to convince his MLS club to cooperate.
However, the source states Gala's board would happily pay 100 per cent of Messi's wages to bring him to Istanbul for the loan stint.