WASHINGTON—A U.S. congressman contacted the White House this week trying to broker a deal that would end WikiLeaks founder Julian Assange’s U.S. legal troubles in exchange for what he described as evidence that Russia wasn’t the source of hacked emails published by the antisecrecy website during the 2016 presidential campaign.

The proposal made by Rep. Dana Rohrabacher (R., Calif.), in a phone call Wednesday with White House Chief of Staff John Kelly, was apparently aimed at resolving the probe of WikiLeaks prompted by Mr....