The maximum stake for FOBTs is scheduled to be reduced from £100 to £2 but this is not set to come into action until October 2019.
Johnson is rumoured to be one of 60 MPs from the Conservative, Labour and Scottish National Parties who are attempting to bring the policy forward to April 2019.
Iain Duncan Smith, ex-Conservative leader, said: "There is firm support for this across the House. The Government has to move."
Johnson previously campaigned against FOBTs in 2015, with the former Mayor of London saying: "They can be dangerously addictive, their promise of whopping wins only gives false hope and they prey on the vulnerable within society."
