Martin reported back for the Sulphurites' first training session of pre-season on Monday, but was absent on Tuesday having travelled to try and finalise terms with an interested party.
"Azza has been off to talk to a National League club today [Tuesday] and I think that's pretty close to being done now," Weaver told the Harrogate Advertiser.
"He came in on Monday for the first day of pre-season but was then away for those talks today.
"There's been a lot of National League interest in Aaron this summer and I'm not surprised because he's a player with a lot to offer."
Martin was transfer-listed alongside fellow striker Danilo Orsi, who recently signed for Grimsby and defensive duo Leon Legge and Nathan Sheron.
And, while Weaver says there have been no approaches for Legge thus far, it appears only a matter of time before Sheron signs for Oldham Athletic.
"Shez has been in talks with a National League club for a while and it looks increasingly likely that he will be finalising that move imminently," the Harrogate chief added.
"It was a really difficult decision to let him go, but you can't keep everyone if you want to evolve the squad.
"He's a lad who I genuinely hope goes and does really, really well for himself in the future."
Town's summer transfer business has seen seven new players arrive at Wetherby Road, while Rory McArdle has agreed a contract extension and academy youngster George Horbury has been handed a professional deal.
That means that Weaver's first-team squad currently numbers 20, with another new face expected to be unveiled on Wednesday.
But, with Martin and Sheron on their way and and Legge also expected to leave, the Sulphurites still have plenty of work to do during this transfer window with at least three more bodies required.