The Premier League champions have identified the Croatia international as one of the priority targets for the summer window and are keen to complete a quick deal with the Bundesliga club.
City have already seen one offer for Gvardiol rejected by Leipzig this month.
But according to Sky Germany, Pep Guardiola’s side have returned with a bid which includes a guarantee of €90m before performance-related bonus payments.
City have also agreed personal terms with the 21-year-old.
The report also claims that Leipzig are already lining up Gvardiol’s replacement and are preparing a move for Lyon’s 20-year-old centre-back Castello Lukeba.
Gvardiol, meanwhile, has attracted strong interest from Chelsea, who submitted their own €90m (£77.4m) bid last summer.
A move to Stamford Bridge failed to materialise but Gvardiol made it clear that he was keen on a switch to Stamford Bridge.
‘I was really confused because one month before the window starts I was upstairs with the sporting director and he says, ‘Yeah Josko, we are not going to sell you, we need you, we believe in you,’ and I was like ok I am fine with this, I can stay here, I like it here, all I need is to work on myself, to play, to win something else. It was like this until the last two days,’ Gvardiol said in an interview with The Times in February.
‘My agent called me and said that Chelsea is extremely interested and, of course, you definitely think about a serious offer from a huge club like Chelsea.
‘Leipzig said they didn’t want to sell me. In the end I really struggled with that decision but we didn’t make an agreement. It is what it is.’
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