The 29-year-old became a free agent at the end of June when his Reds contract expired, bringing an end to an injury-hit six-year stay on Merseyside.
Oxlade-Chamberlain's future is currently in limbo as he searches for his next club, and Villa are believed to be one of several sides offered the chance to sign him.
However, BirminghamLive states that the Lions are set to reject the opportunity to sign the England international, having already brought Moussa Diaby, Youri Tielemans and Pau Torres to the club.
Unai Emery's side are not expected to conclude their transfer business just yet, but the club have no interest in bringing Oxlade-Chamberlain to Villa Park for the 2023-24 campaign.
The ex-Southampton starlet ended his Liverpool career with 18 goals and 15 assists to his name from 146 matches, but he played a mere 13 times during the 2022-23 season for Jurgen Klopp.
Sticking with former Arsenal players, Leicester City are thought to be on the verge of completing a deal to sign Montpellier HSC attacker Stephy Mavididi.
The 25-year-old has made a name for himself in Ligue 1 since departing Juventus' youth setup in 2020, scoring 21 goals in 98 matches for his current club.
Mavididi netted four goals and set up two more in 26 Ligue 1 games for Montpellier during the 2022-23 season, but his contract only runs for another season.
According to the Daily Mail, Leicester are close to finalising a £6.4m for the former England Under-20 international, who is set to undergo his medical in the coming hours.
Mavididi will then officially become the Foxes' fourth signing of the summer, following Harry Winks, Conor Coady and Mads Hermansen to the King Power Stadium.
The 25-year-old failed to make a single first-team appearance for Arsenal between 2011 and 2018, but his tally of 115 Ligue 1 games represents the highest total for an English player in the French top flight.
Heading in the opposite direction to Mavididi, Southampton centre-back Duje Caleta-Car has apparently been offered an escape route back to France by Lyon.
The Croatia international moved to St Mary's in an €8m (£6.9m) deal last summer, and he netted two goals in 19 appearances during his debut season with the Saints.
However, Caleta-Car could not prevent Southampton from dropping down into the Championship with a 20th-placed finish in the Premier League, and he could be part of the club's inevitable mass exodus.
According to the Daily Mail, Lyon have now made contact with Southampton to discuss a possible move for Caleta-Car, whose contract at St Mary's runs for another three years.
Russell Martin's side are unlikely to stand in the 26-year-old's way of a departure for the right price, as they have supposedly earmarked Everton's Mason Holgate as an ideal replacement.
Caleta-Car quickly adapted to life in France after joining Marseille from Red Bull Salzburg in 2018, and he was briefly linked with a switch to Liverpool owing to his displays at the Orange Velodrome.
Finally, the well-travelled James Rodriguez is reported to be closing in on a move to his 10th club, as the 32-year-old prepares to make the switch to Brazilian side Sao Paulo.
Rodriguez has been a free agent since terminating his contract with Greek giants Olympiacos in April, departing with a respectable record of five goals and six assists from 23 games.
Now, Marca reports that the Colombia international is on the brink of making the move to Sao Paulo, who have been engaged in dialogue with his agents for a couple of months.
Rodriguez has apparently given the green light to a contract offer from the Brazilian side, and he will snub interest from MLS teams to return to his home continent.
The report adds that Rafinha's presence at Sao Paulo was a key factor in Rodriguez's decision, as he formed a strong bond with the Brazilian right-back during his two-year loan with Bayern Munich.
Rodriguez has also turned out for Real Madrid, Everton, Monaco, Porto, Al-Rayyan SC, Banfield and Envigado during his distinguished career, winning two Champions League titles with the former.