Mr Boehner has been under pressure from the conservative wing of his party, which has threatened to shut down the government over the funding of women's health group Planned Parenthood.
An aide warned of "prolonged leadership turmoil" if he stayed in office.
The news comes a day after Mr Boehner hosted Pope Francis at the US Congress.
The 65-year-old was expected to make a public announcement at 10:00 local time (15:00 BST).
But after a long delay, he failed to appear and left the Capitol building through a backdoor without comment.
He assumed the leadership position in January 2011, when Republicans took control of the House. However, since then he has faced down several rebellions.
In a statement, one of his aides said: "The Speaker believes putting members through prolonged leadership turmoil would do irreparable damage to the institution."