The pair have spent the majority of the summer together on international duty at Euro 2024 in Germany, where England were looking for European glory.
Both Palmer and Watkins failed to start a single game in the competition but still managed to have a major impact on proceedings off the bench.
Villa star Watkins bagged the last-minute winner for the Three Lions in the semi-final victory over Netherlands at Signal Iduna Park last week.
Chelsea man Palmer came off the substitutes' bench in the closing stages of the final against Spain and netted an equaliser during the final 20 minutes.
Chelsea's Cole Palmer 'convincing' Watkins to leave Aston Villa
According to Football Insider, Watkins is being told by a certain England teammate that he should end his four-year association with Aston Villa this summer.
The report claims that Chelsea's Palmer has been convincing the former Exeter City man to depart from the Second City club and join the Blues.
It is understood that the pair have grown close while on international duty in Germany this summer, with Palmer assisting Watkins for his strike against the Netherlands.
Palmer supposedly believes that himself and the Villa marksman could take Enzo Maresca's side to the next level following the departure of Mauricio Pochettino.
As well as the West Londoners, it is also believed that Arsenal are keen on a move for Watkins, who bagged 19 goals in the Premier League last season.
Chelsea to offer player swap for Watkins?
Chelsea would allegedly be willing to offer homegrown talent Conor Gallagher to Villa in a bid to lure Watkins to Stamford Bridge.
The future of the England midfielder is in doubt amid a busy summer window, with Tottenham Hotspur also known to be interested in the former Crystal Palace loanee.
Villa's need for Gallagher has to be in question, though, considering that Unai Emery's side are said to be close to the signing of Everton's Amadou Onana.