The potential cash-and-stock deal, which values Andeavor at about $150 per share, is expected to be announced on Monday, the WSJ reported, citing sources.
The offer would represent a premium of 22.6 percent to Andeavor stock's Friday close.
The deal is expected to produce $1 billion of synergies, the paper said.
Marathon Chief Executive Gary Heminger is expected to run the combined company, with a senior role for Andeavor's chief executive, Gregory Goff, it said.
San Antonio, Texas-based Andeavor, formerly known as Tesoro, operates 10 refineries in the western United States with a refining capacity of about 1.2 million barrels per day, and ownership in a logistics business, according to Andeavor website.
Marathon and Andeavor were not immediately available for comment outside regular business hours.