WASHINGTON—Former national security adviserMike Flynn will turn over documents from his businesses to the Senate Intelligence Committee, according to a person familiar with the matter, easing the possibility of a protracted legal standoff over his cooperation with the panel’s investigation.

According to this person, Mr. Flynn will start to turn over records demanded by Senate investigators from two companies that he controlled, as well as some personal documents as soon as next week. Those records had been subpoenaed by the...