On Tuesday, news headlines suggested Lewis-Skelly could be on the move with two Premier League clubs reportedly interested.
One source told BBC Sport that Lewis-Skelly was offered to Manchester United in January, but Arsenal say they have not actively offered him to anyone.
Manchester United do need a left-back though, with their most recent signing in the position, Patrick Dorgu, playing in an advanced role throughout this pre-season.
Dorgu was mostly injured after Michael Carrick took charge but all of his Premier League appearances under the United boss were on the wing or in a wide attacking role.
Luke Shaw started every Premier League game last season but his overall fitness record is patchy.
United have been linked with a move for Newcastle’s Lewis Hall, but accept it would be difficult to sign him, particularly after the departures of Sandro Tonali and Guimaraes from the club.
If they did sign Lewis-Skelly there is also the option that he could fill the third midfielder position there is still debate over, but the feeling is that United want another natural full-back to challenge for a starting role.
Like United, Chelsea also say they have been offered the chance to sign Lewis-Skelly but Arsenal maintain they have not offered the player to any other team.
Chelsea have two left-backs already in Jorrel Hato and new signing Pep Chavarria and are not in Europe, so will not have as many fixtures to navigate this season.
With that in mind they are not expected to sign a third left-back to supplement their squad. Chelsea also have six central midfielders, although it is possible Enzo Fernandez could be sold and Dario Essugo leaves on loan.
It makes it unlikely Chelsea would be in the market to sign Lewis-Skelly without substantial sales.