The 29-year-old has spent five years at Anfield, taking in 170 appearances for the Reds.
Lovren has, however, seen his position in Jurgen Klopp’s plans come under threat.
The emergence of Joe Gomez and form of Virgil van Dijk and Joel Matip has seen him slip down the defensive pecking order.
Barnes believes his experience could serve Liverpool well heading forward, but concedes that personal ambition may lead Lovren away from Merseyside.
The Liverpool legend told The Sport Review: “With Joe Gomez coming back, if Lovren feels that he’s not going to be playing anymore and he wants to play first-team football, fine.
“I quite like him at Liverpool because, of course, if players get injured, he can step in and do a good job.
“But players sometimes think to themselves that they want to play first-team football so therefore if he leaves, I’ll have every respect for him, rather than him just staying at Liverpool in the comfort zone.
“Maybe it is time for him to move on if that’s what he decides.”
Serie A is considered to be the most likely destination for Lovren if he does move on this summer.
AC Milan are in the process of piecing together another squad refresh, with their managerial reins having passed from Gennaro Gattuso to Marco Giampaolo.
It remains to be seen whether Lovren remains a target for the new man at the helm.
The Croatian is, however, considered by many to be disposable for Liverpool, with former Reds star Steve Nicol having previously told ESPN FC: “No [I would not keep him]. You have Gomez, you have Matip.
“Lovren’s been a good servant to Liverpool.
“Just because on occasions he’s made horrendous mistakes and we’ve picked him up on it, it doesn’t meant that I don’t like the guy.
“The guy’s got a big heart and he’s given everything for Liverpool. He’s played well on occasions.”