The Blues have a long-term interest in the midfielder and have made another check on his availability following Arsenal's signing of Bruno Guimaraes.
However, Arsenal do not want to sell Lewis-Skelly to a local rival and a team they may have to compete with for the top places this season. Chelsea were linked with Lewis-Skelly in January.
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Manchester United also have a long-term interest in Lewis-Skelly. However, United are still deciding whether they will commit to signing another midfielder or a left-back before the deadline.
It is questionable whether United would be able to do any more significant deals should they make a move for Lewis-Skelly.
This is amid a market of high prices for central-midfield players this summer, although he can play both positions.
Last season, Arsenal kept their squad together despite a big spend. They want to be ambitious again this summer - but there is an acceptance there will have to be sales.
Players like Ethan Nwaneri and Lewis-Skelly would represent pure profit.
More to follow.
