The Sun says Palace are ready to offer £10million for Gallagher.
Manager Roy Hodgson sees the 20-year-old as an ideal prospect as he tries to strengthen his squad.
He hopes Blues boss Frank Lampard will allow Gallagher to leave permanently after loan spells at Charlton and Swansea this season.
Gallagher — who has two years to run on his Stamford Bridge deal — has shown his potential at Championship level with his strong running and goal threat.
He has represented England at all levels from Under-17 to Under-21.