The 60-year-old took over in January in a caretaker capacity when Roy Hodgson was sacked just six months into the job.
Although his appointment was seen as a stop-gap by the owners, Dalglish has exuded the air of a man comfortable in the role he last occupied in 1991 and results on the pitch have been encouraging.
Talks over a new deal have apparently been under way for three weeks but no timescale has been put on negotiations.
"We have a taken a wait and see approach," Fenway Sports Group president Sam Kennedy told BBC News. "The conversations are happening now with the appropriate people - Ian Ayre (managing director), Damien Comolli (director of football) and the ownership.
"But I do think he (Dalglish) has provided a great lift. He is a wonderful man, he is passionate about the club."
Source: PA
