The Canadian singer and the 'Mean Girls' star were both at popular Los Angeles hotel Chateau Marmont when the argument began, with Avril branding Lindsay a "fake".
A source said: "Avril was at a table with friends just over a week ago including her boyfriend, Brody Jenner, when Lindsay came over to say hi.
"But as soon as she approached, Avril launched at her and said, 'Get the hell out of my face, you are fake, you are a loser. I don't like false people. Stay away from me and my friends.'
"Lindsay was furious and screamed back, 'Don't threaten me!' She then stormed off to security and tried to get them to kick Avril out. They refused to force Avril to leave. Lindsay eventually stormed off."
The pair are said to have fallen out because Lindsay snubbed the 'Girlfriend' hitmaker at a showbiz event but tried to make friends with her when she saw she was hanging out with people she was eager to impress.
The source added to the New York Post newspaper: "It seemed like she was just being friendly to Avril because she was with people Lindsay wanted to impress. Avril is avoiding going to the Chateau Marmont when she knows Lindsay will be there. She doesn't want anymore drama."
It is the second time this month Lindsay's behaviour at the hotel has caused concern to onlookers.
Last week, the 23-year-old star partied with her mother Dina until the early hours of the morning at the Chateau Marmont, with fellow revellers proclaiming she looked "pale and bloated".
A source said: "Lindsay didn't look good. Her face is so puffy and she seemed exhausted. And everyone around is just looking at Dina like, 'Why the hell are you partying with your daughter?' Doesn't she see what she's doing and how weird and wrong it is?"