Croatian centre-back Lovren enjoyed an excellent debut season on the south coast, but has emerged as a prime target of Rodgers, who is monitoring several possible recruits for the heart of his defence.
Lovren cost £8.5 million when he joined the Saints last season and his valuation will have significantly increased.
Liverpool have made Southampton midfielder Adam Lallana a priority signing as Mauricio Pochettino’s side are in danger of being broken up over the summer. However, with a move for Lallana stalling, Liverpool have a back-up plan to line up Swiss international Shaqiri.
The 22-year-old midfielder has been scouted extensively by Liverpool since his Basle days and, at around £10 million, would be considerably cheaper than Lallana.
Rodgers will also revive his pursuit of Chelsea full-back Bertrand as he seeks to find a solution at left-back.
Part of the problem in negotiating for Bertrand is Chelsea may not be willing to sell to a top-four rival. There is also bad blood between the clubs following Liverpool’s loan capture of Victor Moses, who they barely used.
Bertrand, 24, who spent last season on loan at Aston Villa, is Rodgers’ preferred choice at left-back, although Seville’s Alberto Moreno has also been extensively scouted.
Rodgers has made it clear he wants well-established players added to his squad, including a striker to add competition with Daniel Sturridge and Luis Suárez.
One of those under consideration is Swansea’s Wilfried Bony, who - like Lovren - enjoyed an impressive first season in the Premier League.
Bony would command a fee in excess of £15 million.
Meanwhile, Rodgers says there is no possibility of his captain Steven Gerrard dwelling on his Premier League disappointment when the World Cup starts.
“I have no concerns whatsoever,” said Rodgers.
“I think Steven will go away and when he reflects on what has been personally a brilliant season of football, he has been very unfairly pointed out in terms of his slip. It could have happened in the third game, it could have happened in the fourth game ,but it certainly wasn’t anywhere near the defining moment of our season. That was not what cost us the league.
"It was more games when we won the games. When we won 5-3 at Stoke when we conceded, when we won 6-3 at Cardiff, when we won 4-3 at home and conceded three. It was those games, not Steven’s slip or Kolo’s [Toure] misplaced pass [at West Brom]. Because he [Steven] is a very conscientious boy and he loves Liverpool and he is very hard on himself.
"He’s his own worst critic but he comes away recognising that he has been involved in a real title run-in and next season he will be better for it and we will be a better squad for it and we will be ready to fight again and he will play a pivotal part in it. He will take that confidence into a world cup.
“I always say to players when you go to England don’t think of Liverpool. Think about your country and the team you are playing for.
"One of his qualities is that he can do that. He is a model professional at that, he is very proud to play for England and captain his country and I am sure he will go and do the country proud again.”
