The bid was rejected but Lovren said his "head is already at Liverpool".
Saints have seen Rickie Lambert, Adam Lallana and Luke Shaw, plus coach Mauricio Pochettino, leave this summer.
"Look, I'd gladly stay in Southampton if the club had any ambitions, if they kept the key players," Lovren told Croatian paper Sportske Novosti.
The player, whose agent told the BBC the quotes were accurate, added that "not even the Liverpool bid" would have persuaded him to leave had England internationals Lambert, Lallana and Shaw stayed.
Striker Lambert and midfielder Lallana have already completed moves to Liverpool, while defender Shaw has joined Manchester United.
Pochettino has also left to take over at Tottenham, while chairman Nicola Cortese, who brought Lovren to St Mary's, left in January.
"When I decided to go to Southampton, many people were surprised, but I went because the people at the club had great ambition," said Lovren.
"It was because of that I'd agreed to sign for Southampton. When the chairman left, and Pochettino was going too as early as April, I realised things had fallen apart."
The defender said he was also upset by the way Southampton have handled the Liverpool bid.
"Liverpool sent the offer and the club haven't informed me," he said. "Not even about the subsequent Liverpool bids. That wasn't right.
"I found out about the bid from other people which disappointed me and I realised I've got no business staying at Southampton. Frankly, my head is already at Liverpool."
Lovren, who was part of the Croatia team that finished third in Group A at the 2014 Fifa World Cup in Brazil. joined the Saints from Lyon for £8.5m last summer. He is currently 12 months into a four-year contract.
Southampton centre-back Dejan Lovren says he wants to move to Liverpool, who offered £20m for the 25-year-old Croatia defender earlier this week.