With the Croatia international having dropped down the pecking order at Anfield, it has been suggested that the centre-back could be sold during the next transfer window.
While that remains the case, Sky Sports News claims that the 31-year-old may be provided with a new deal.
The Premier League champions have the option of extending Lovren's deal until 2022, something which would theoretically retain his value in the transfer market.
Zenit St Petersburg are said to be monitoring the situation ahead of a possible approach later in 2020.
Lovren has made just 15 appearances in all competitions during the current campaign.