The former defender also says that the summer's business is about more than just signing players.
The manager recently attended the end-of-season review with the club’s owners Fenway Sports Group after a disappointing campaign saw them finish sixth in the table, with a heavy 6-1 defeat to Stoke on the final day putting pressure on him.
He was, however, backed by the owners although long-serving assistant Colin Pascoe and first-team coach Mike Marsh have left the club, according to the Telegraph.
Despite the backing from the board, Lawrenson believes the next six months will be crucial to ascertaining if Rodgers will be at the club for the long-term.
“Fenway Sports Group have made the right decision by sticking with Brendan Rodgers,” he said.
“[But] the next six months will be crucial. If things go well, FSG will look to back Rodgers in the January transfer market. If not, it could well be a sign the manager is on his way.”
The Reds are expected to be busy during the summer transfer window, with a striker thought to be top of their wish list, following the struggles of Mario Balotelli last season.
They have already secured the signing of James Milner on a free transfer from Manchester City, and are being heavily linked with a number of other players.
Lawrenson though, says there will be more to Rodgers' summer agenda than making acquisitions.
“This summer's tasks aren't just about signing new players,” he added.
“It'll be about improving the players that he brought in last year, and getting rid of the deadwood too. That will have been part of the conversation. Talk, though, is cheap. And it's now up to Rodgers to put words into action.”