The former Boro defender parted company with the club on Monday evening as his near three-year reign came to an end with the Teessiders sitting in 16th place in the table.
However, while budgets have been drastically reduced since Boro slipped out of the Premier League at the end of the 2008/09 season, Gibson is adamant the cash has always been there to make a fist of promotion.
He told the Evening Gazette: "Financial restrictions? We all have financial restrictions, yet it seems to be used as an issue to kick us when we are down.
"Every organisation has financial restrictions, from Manchester United to Aldershot.
"What we have strived to do as a football club is to make sure our managers have had the means to compete, whether it's in the Premier League or the Championship.
"I am satisfied we have always done that, whether it's Steve McClaren or Tony Mowbray, and by doing that, you give your manager the resources and the responsibility to compete."
Mowbray's number two Mark Venus has been placed in temporary charge of team affairs, and it is understood Gibson is yet to talk to potential candidates as he assesses his options.
Ian Holloway and Mike Phelan currently head the bookmakers' lists, with Stuart Pearce, Alex McLeish and Martin O'Neill also figuring prominently in the odds.
Watch Middlesbrough take on Doncaster in their first post-Mowbray match against Doncaster Rovers live on Sky Sports 1 HD at 7.30pm on Friday