The 29-year-old spent most of last season on loan at Huddersfield, where he scored nine goals in 22 appearances.
Leicester paid Everton £3m for Beckford in August 2011 but they are prepared to sell him on and Gartside confirmed he is set to move to the Reebok Stadium.
"We have got one more at least, who we will announce very shortly, then it is a case of settling down and see where we go for the first few weeks of the season," he told BBC Radio Manchester.
Gartside went on to confirm the player in question was Beckford.
"We had a bit of a hiccup on Jermaine Beckford but we have probably sorted it out now," he said. "Jermaine will join us in the next couple of days."
Bolton finished seventh in the Championship last season, missing out to Leicester for the final play-off spot after a strong finish to the campaign, but they are among the early favourites for promotion in 2013/14.