Pricing remains in flux and no written offer is on the table, sources said on Thursday.
According to the Times, the offer amounts to $4.7 billion (€3.9 billion), slightly less than a five-billion-dollar bid by Sony.
Sony's offer, which includes debt assumption, would allow the Japanese consumer electronics, movie and music giant to acquire 80-year-old MGM studio's 4,200-title library of post-1948 movies and create the industry's largest collection.
Time Warner's bid would pay about $13 in cash to MGM shareholders and about $11.50 dollars a share in Time Warner stock to the studio's major shareholder, billionaire Kirk Kerkorian.
The Times report said that Kerkorian was particularly attracted by the idea of becoming a major shareholder in Time Warner.
At MGM's shareholder meeting on Tuesday, studio executives told investors MGM was exploring alternatives. There was no mention of a pending Sony deal.
Shares of Time Warner ended on Wednesday's session up one percent at $17.58 dollars. MGM shares shed one percent to close at $12.10 dollars. Sony shares added nine cents to finish the session at $38.05 dollars.