Peter Box said the organisation was sifting through applications for the role and the successor to Sir Gary, who resigned on health grounds in March in the midst of bullying and expenses allegations, would be appointed "before the end of this month".
In October, it was revealed almost £50,000 had been spent by WTY so far on its efforts to recruit a new chief executive.
Coun Box said: "I think there are candidates who look really, really good so I'm very positive that an appointment can be made who can take the organisation forward to its next stage.
"And the next stage is combining the best of what Welcome to Yorkshire has to offer with perhaps the best in terms of governance."
Coun Box took over as chairman last month, announcing days later that he would be stepping down as leader of Wakefield Council after 21 years at the end of November.
He has spoken of a "spend now, worry about it later" culture pushing the agency to the brink of collapse.
The company, which is privately-run but receives millions in public funding, has been struggling to recover from the fallout to the resignation of its ex-chief executive Sir Gary, with chairman Ron McMillan resigning in April and interim chairman Keith Stewart ousted by council leaders last month before Coun Box's appointment.
Subsequent inquiries found that Sir Gary’s behaviour towards staff had “fallen short” of expected standards, while it was found he had claimed around £26,000 in “personal” expenses “not incurred wholly for the benefit of Welcome to Yorkshire”.
Investigators were unable to determine whether a further £900,000 of expense claims by Sir Gary and other senior officials had been “reasonable” because of a lack of clear spending policies at the organisation.
Coun Box said: "Once we've got a chief executive in post, I think that we can start to regain public trust at a faster rate."
The Yorkshire Post has previously revealed WTY would have run out of money and been unable to pay staff in September if it were not for a £500k loan from North Yorkshire County Council and a further £1m in support from North and West Yorkshire councils in October. The latter was agreed on the condition that interim chairman Keith Stewart was replaced.
After taking over as chairman last month, Coun Box ordered the publication of two previously-classified reports into the company's financial position, with some details partially redacted, following a promise to make WTY's operations more open and transparent.
A press release was also issued which showed costs surrounding the fallout from Sir Gary's departure totalled nearly £500k and included investigation costs, CEO recruitment, legal costs and payments in lieu of notice.
Kirklees Council chief executive Jacqui Gedman is currently providing the organisation with "strategic support".