Double world champion Alonso had tried to pass Magnussen's Haas round the outside of the first chicane during the second phase of qualifying at Monza and then made light of the incident over team radio.
"Magnussen wanted to race into Turn One," he told his engineer.
The Dane, who failed to progress to the final top 10 shootout, was not amused.
"I'm not going to let him pass me, and sacrifice my own lap. No way. I know he thinks he's a god, but no way," he told reporters.
"Obviously he got a perfect slipstream and thought he could overtake into Turn One -- but I'd rather hang myself," added the 25-year-old.
"He came to me after qualifying and laughed to my face. Just outright disrespectful. Can't wait for him to retire."
Alonso, who won his titles with Renault, has announced he is leaving Formula One at the end of the season and is expected to turn his focus on racing in America in a bid to win the Indianapolis 500.
The Spaniard, who has endured an uncompetitive car for years with McLaren, has complained that Formula One has become predictable and boring.
Magnussen suggested Alonso was good at talking up his own ability to get the most out of a car that his team mate Stoffel Vandoorne has struggled with.
"He's talking about his laps being divine and whatnot. He literally thinks he's a god. It's pretty amusing," commented the Dane.
Asked about the comments at a later McLaren media briefing, Alonso had nothing to add.
Both men were summoned to see stewards after the track incident but no further action was deemed necessary.
(Reporting by Alan Baldwin, editing by Pritha Sarkar)
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