David Woods of the Daily Star exclusively reported as much, with the Blues hoping to test the Gunners' resolve on a player who has yet to agree terms on a new deal despite being in the final year of his contract.
The midfielder is said to be receptive to a switch to Stamford Bridge, where he can play UEFA Champions League football and has a better chance of seeing his demands for £120,000 per week met.
Arsenal boss Arsene Wenger will not want to part with any of his stars this summer, particularly to direct rivals, and he recently spoke highly of Oxlade-Chamberlain, as relayed by the Evening Standard's James Olley:
The England international has yet to fulfil his potential, but he only turns 24 on Tuesday, so while he may not be considered a youngster anymore, he's still got the best years of his career ahead of him.
He had a productive 2016/17 campaign, notching six goals and 11 assists. His versatility would be of great use to the Blues.
Oxlade-Chamberlain could provide cover in either wing-back position, in central midfield or even in one of the forward positions behind Alvaro Morata.
Bleacher Report's James McNicholas noted the variety of positions the England international was tasked with adopting in pre-season:
He's also a force to be reckoned with when running with the ball at his feet, per Squawka Football:
A move to Chelsea could get the best out of him, as his career at the Emirates Stadium has perhaps gone a little stale despite his contributions last year.
He'd make for a strong buy, but Arsenal will be particularly reluctant to allow him to leave for Chelsea, even with a substantial offer on the table.
Meanwhile, Newcastle United are reportedly closing in on a season-long loan deal for Chelsea's Kenedy.
According to Neil Moxley, writing for the Mirror, Magpies manager Rafa Benitez, formerly the boss at Chelsea "has moved quickly to secure the deal using well-established contacts made during his spell in charge of the club four years ago."
The Chronicle's Lee Ryder gave a further update:
Newcastle have already brought in Jacob Murphy as a wide option alongside Matt Ritchie and Christian Atsu, whom they signed on a permanent deal from Chelsea earlier in the summer, having had him on loan last season.
The Magpies are crying out for reinforcements at left-back, though, as neither Paul Dummett nor Massadio Haidara has particularly convinced, so Kenedy might also provide competition there.
The 21-year-old struggled for form and fitness on loan at Watford last year and made just three appearances, but if Benitez can get the best out of him, he could be a useful addition to a side in need of more quality.
What's more, he's a low-risk acquisition on loan, which would suit the promoted side if he struggles to impress.