Guardiola is intent on overhauling his entire defence after a trophy-less 2016/17 campaign, and has already agreed a £35m deal to sign Brazilian goalkeeper Ederson Moraes from Benfica.
The Spaniard intends to partner Van Dijk with John Stones in central defence next season and is expected to make moves for Tottenham right-back Kyle Walker and Monaco left-back Benjamin Mendy to fill the full-back positions.
The Times says Southampton are holding out for a transfer fee of £60m for Van Dijk as they are obligated to pay 10% of any fee they get for the defender to Celtic. The Saints signed the 25-year-old from the Scottish club for £13m in 2015.
City's offer for the Netherlands international is understood to be higher than what Chelsea and Liverpool are prepared to pay – the Blues are reportedly prepared to offer him £170,000 a week, while the Reds are prepared to pay around £120,000 per week.
Southampton boss Claude Puel has insisted that Van Dijk would stay at St Mary's next season, pointing out that he 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."
City have already signed midfielder Bernardo Silva from Monaco in a £43m deal. Arsenal forward Alexis Sanchez is reported to among Guardiola's other transfer targets for the summer.