Smith, 25, has penned a two-year deal with City after rejecting a one-year contract extension at The County Ground. He scored eleven goals last season.
Bristol City poach Swindon duo
Heywood, also 25, has agreed a three-year deal at Ashton Gate and is set to sign later this week.
Former Sunderland striker Marcus Stewart, 31, also looks set to sign before the end of the week too.
City missed out on the playoffs by one point last season and manager Brian Tinnion is looking to strengthen a side who are one of the favourites for promotion to the Championship.