WASHINGTON—The Senate Intelligence Committee has sent letters to at least four former associates of the Trump campaign asking for information and records about their activities before and after the 2016 election, a sign that the committee’s probe into alleged Russian campaign interference is heating up.

The committee, which carries subpoena power, sent letters to former Trump campaign Chairman Paul Manafort, former national-security adviser Mike Flynn, former campaign adviser Roger Stone and former campaign foreign-policy adviser Carter Page, according to people familiar with the matter.