Steve Bannon has vowed to lead a “season of war” on Washington’s Republican establishment. But donors most likely to fund a disruptive campaign may be hard to win over.

Mr. Bannon, the ex-White House strategist and Breitbart News executive, is planning to support primary challengers to GOP senators with a goal of toppling Senate Majority Leader Mitch McConnell, a lawmaker that in recent speeches he has called ineffective and insufficiently loyal to President Donald Trump. Mr. Bannon, who hasn’t ramped up any fundraising yet, has said he wants to defeat five GOP senators up for re-election next year.

Turning that into reality likely would require tens of millions of dollars to rally voters and build the name recognition of various candidates facing well-financed, sitting senators. Indeed, outside groups spent nearly $3 million in 2012 helping Richard Mourdock defeat Republican Sen. Dick Lugar in Indiana. Mr. Mourdock went on to lose the general election.

In recent weeks, however, two of the biggest GOP donors who have funded disruptive campaigns in the past have distanced themselves from Mr. Bannon. And the turbulent Senate race in Alabama, where Mr. Bannon backs Republican candidate Roy Moore, who is facing sexual assault allegations, has energized Mr. McConnell’s defenders.

Sheldon Adelson, a Las Vegas gambling billionaire, and his wife indicated this week that they won’t be part of Mr. Bannon’s election efforts. “They are supporting Mitch McConnell 100%. For anyone to infer anything otherwise is wrong,” Mr. Adelson’s spokesman, Andy Abboud, said in a rare statement of the Adelsons’ views.