The Blues are believed to be looking for a suitable offer to sell Gallagher this summer in an attempt to meet the Premier League's Profit and Sustainability Rules.
Selling the England international would represent pure profit for the West Londoners, given that the midfielder came through the club's academy.
Enzo Maresca's side have already added five new players this summer, including Kiernan Dewsbury-Hall, Omari Kellyman, Renato Veiga, Marc Guiu and Tosin Adarabioyo.
However, a report this week suggests that the Blues are still looking to strengthen their squad further before the new Premier League season begins, with a central midfielder being the main priority on their list.
Blues to make shock bid for Fulham's Andreas Pereira?
According to Globo, Chelsea have initiated talks with Pereira's agent, Kia Joorabchian, regarding the potential transfer of the Brazil international to Stamford Bridge ahead of the new season.
The report adds that Maresca has personally endorsed Chelsea's efforts to sign the 28-year-old, and the club is prepared to offer up to £35m to Fulham for the transfer.
It is understood that Pereira received a verbal agreement from his current employers allowing him to leave Craven Cottage this summer if an acceptable offer is made.
Monaco, who secured second place in Ligue 1 last season, earning a spot in the revamped Champions League this term in the process, are also believed to have signalled their interest in the Brazilian midfielder.
However, Pereira reportedly prefers to stay in the Premier League rather than move to France, and a switch to Stamford Bridge would also allow him to remain in London.
The former Man United player came off the bench in three matches for Brazil at the Copa America this summer and has remained in the US on holiday since.
He is expected to return to England and join the Cottagers at the end of the month for the start of their pre-season training camp in Portugal.
Chelsea to continue clearout this summer
While Chelsea have already brought in five new players this summer, they have also managed to get a few significant wage costs off their books.
Among the players who have departed Stamford Bridge in the offseason are Thiago Silva, Hakim Ziyech, Lewis Hall, Omari Hutchinson and Ian Maatsen.
However, it is understood that additional exits are to be expected in the coming weeks, with the club still looking to bring in a top-level striker to support Nicolas Jackson, who had a mixed first season in West London.
Along with Gallagher, fellow academy graduates Trevor Chalobah and Armando Broja are believed to be up for sale this summer.
There have been several reports in recent days suggesting that Atletico Madrid, Tottenham Hotspur and Aston Villa have all signalled interest in Gallagher, while Chalobah is on the radar of Crystal Palace.
Broja, on the other hand, was linked with clubs in Serie A earlier in the transfer window, but Southampton have now emerged as a strong candidate to sign the Albanian.
Still only 22 years of age, Broja spent the duration of the 2021-22 Premier League season at St Mary's and scored nine goals under Ralph Hasenhuttl, who was eager to secure his services on a permanent basis back then.