The Daily Mirror says Bertrand, 27, has told his friends that he expects to leave Southampton after reports emerged earlier in the week indicating that Van Dijk was close to a £60m ($77m) move to Liverpool.
City boss Pep Guardiola has been tracking Bertrand for months and could move for the England international if he fails in his pursuit of Monaco left-back Benjamin Mendy.
Guardiola has made the signing of a left-back a top priority after releasing Gael Clichy at the end of the 2016/17 season.
Chelsea are also interested in re-signing Bertrand, whom they sold to Southampton for a reported fee of £10m in 2015.
Liverpool boss Jurgen Klopp is in the market for a left-back and is understood to be an admirer of Bertrand, but a transfer is complicated as the Reds are also trying to sign Van Dijk.
Southampton are unhappy with the Reds' interest in Van Dijk and are set to report the Merseyside club to the Premier League over an alleged illegal approach for the centre-back.
Chelsea and City have also been linked with a move for Van Dijk, but the defender is understood to prefer a move to Liverpool as he wants to work with Klopp.
Saints boss Claude Puel has insisted that Van Dijk would stay at St Mary's next season, pointing out that the Dutchman was contracted to the club until 2022.
"He's our captain and he's important for us. He's matured and taken leadership on the team; he's a fantastic player we know well," Puel was quoted as saying by the Daily Telegraph in May.
"He's important for us and now, of course, we will wait to the beginning of the season to see him on the pitch with Southampton. He has a long contract and the directors will want to keep him."