ESPN's Transfer Talk column reports that both London rivals are competing to sign the Bundesliga club's attacking midfielder.
Monday's transfer round-up includes Barcelona monitoring Chelsea forward Estevao, interest in Erling Haaland, and a possible European return for Joao Felix.
Several Premier League clubs are said to be watching Trent Alexander-Arnold's situation, according to transfer reports.
Sunday's football rumour mill also links Trent Alexander-Arnold's continued stay at Real Madrid and Premier League interest in Spurs' Micky van de Ven.
Saturday's football rumour mill also links Premier League clubs to Atletico Madrid's Marcos Llorente and Juventus to Crystal Palace's Tyrick Mitchell, according to BBC Sport.
The latest round of transfer speculation links Chelsea with a move for Bournemouth midfielder Alex Scott, while Jude Bellingham has dismissed reports of Manchester City interest.
Reports link Bayern Munich attacking midfielder Lennart Karl with a possible transfer to Real Madrid, according to ESPN's Transfer Talk coverage.
Reports link Liverpool to keeping Florian Wirtz while Chelsea eyes an English midfield pairing, among Thursday’s gossip column items.
Manchester United and Liverpool are reportedly interested in Crystal Palace defender Tyrick Mitchell, while La Liga clubs are said to be watching Aston Villa loanee Alejandro Garnacho and Chelsea are linked with Gambian teenager Adama Jeng.
Liverpool could bid for Crystal Palace's Adam Wharton as soon as January, according to ESPN.
Manchester City and Liverpool are also said to be eyeing Real Sociedad defender Jon Aramburu, according to ESPN.
The Premier League club is said to be considering the versatile attacker as a potential reinforcement, according to ESPN.
The reported interest comes alongside unresolved questions over JJ Gabriel's next step and Antonio Conte's future plans, according to BBC Sport.
ESPN reports United are seeking to retain their captain as discussions on an extension face early hurdles.
BBC Sport's Sunday gossip column reports interest from Arsenal in the struggling Brazilian forward, alongside claims of a possible Barcelona bid for Erling Haaland and Liverpool's stance on Florian Wirtz.
ESPN's Transfer Talk column reports that both Spanish giants could pursue a future move for the young American midfielder.
ESPN's Transfer Talk column reports that winger Noni Madueke could leave Arsenal in January amid a search for more playing time.
BBC Sport's Friday gossip column reports interest from Arsenal, Real Madrid and Chelsea in a trio of players ahead of the next transfer window.
BBC Sport reports the Spanish club has shown significant interest in signing the teenage Manchester United forward.
ESPN's transfer roundup says LaLiga giants Barcelona and Real Madrid are eyeing a potential move for the Manchester City striker.