The 26-year-old forward is currently in talks with a number of interested parties and looks set to depart Wetherby Road less than a year after signing from National League Maidenhead on a two-year deal.
"I had to be frank with him and tell him that both Luke [Armstrong] and Jack [Muldoon] are ahead of him to start games.
"He’s had a year of not playing too many matches with both us and Boreham Wood, so he’s in agreement with me. He desperately wants to start games."
Orsi was not initially named on the League Two Sulphurites' transfer list alongside fellow attacker Aaron Martin and defenders Leon Legge and Nathan Sheron when Town revealed which members of their squad were being released and retained last month.
But Weaver has explained that his omission was due to the fact that at that point, Orsi and loan club Boreham Wood's season was still not finished.
"I didn't think it was right to have that conversation with Danilo before Boreham Wood were done for the summer," he revealed.
"They still had points to play for and I didn't want me breaking the news to affect Danilo in the dressing room."
Prior to joining Boreham Wood on loan in January, Orsi made 14 competitive appearances for Town during the first half of 2021/22.
And he will leave North Yorkshire boasting an impressive goal-scoring record, having found the back of the net on six occasions.
He opened his account for Harrogate with an EFL Trophy hat-trick in a 3-1 success over Mansfield Town before going on to register against Newcastle United Under-21s in the same competition, then hitting a late winner during November's 2-1 FA Cup victory at home to Wrexham.
His solitary League Two strike for the club came from the penalty-spot as the Sulphurites thrashed whipping boys Scunthorpe United 6-1 on October 9, but he did only start three league matches.
His spell in the National League with the Wood was however less impressive and ended with him having contributed just two goals in 24 matches.
With both Orsi and the versatile Sheron close to agreeing moves away from Wetherby Road, their exits would leave Town with just 12 senior players on the books.
But, with Martin and Legge no longer in Weaver's plans, the Sulphurites are certainly short on numbers ahead of the 2022/23 campaign, which is scheduled to get underway on July 30.
Weaver did reveal last week that he is in "advanced talks" with five transfer targets and has subsequently confirmed that he expects to be able to unveil a second new signing - following the capture of goalkeeper Pete Jameson - before the weekend.