The Russia investigation that Donald Trump’s legal team predicted would clear the president by year’s end looks to stretch into 2018, prompting his supporters to ratchet up complaints that Special Counsel Robert Mueller’s operation is politically motivated.

The calls for a more aggressive approach have intensified amid disclosures that a senior agent on Mr. Mueller’s team, Peter Strzok, had sent text messages that were allegedly critical of Mr. Trump during the 2016 election. Mr. Mueller subsequently removed Mr. Strzok from the probe.

Republicans also point to Andrew Weissmann—a Mueller deputy who had applauded the Justice Department’s decision not to defend the initial White House travel ban on people from majority Muslim nations—as evidence of bias on the special counsel team.

The president’s legal team has largely stayed quiet on the issue. But with Mr. Mueller’s investigation appearing to edge closer to Mr. Trump’s family and inner circle, the president is being urged to drop his team’s mostly cooperative approach to date.

“The president’s lawyers are sleepwalking their client into the abyss,” said Roger Stone, an adviser to Mr. Trump’s 2016 campaign whose dealings with WikiLeaks are under examination as part of the Russia investigation. “They are entirely unrealistic about the enmity toward the president from the political establishment, and the established order” Mr. Stone added.