The Bantams turned down a request from Sheffield United to speak with Parkinson last week about the current vacancy at Bramall Lane.
Parkinson has 12 months remaining on a three-year deal and his future was in doubt as the Blades’ approach coincided with Italian businessman Gianni Paladini’s takeover bid.
But it is understood the Blades were not willing to activate a reported six-figure get-out clause in Parkinson’s contract and turned their attentions elsewhere.
The South Yorkshire club are expected to unveil Nigel Adkins as their new manager at a press conference in London at 5pm on Tuesday.
“As far as I’m concerned we turned down an approach from Sheffield United. We wanted Phil to stay,” Bradford’s joint chairman Mark Lawn said.
“Phil wanted that to happen, it has happened and we will be talking with Phil further.”
Paladini, who first approached Bradford in February, failed in his bid to complete a deal before his exclusivity agreement expired at midnight last Friday.
It is understood Paladini’s bid collapsed because one of his major backers pulled out, but the former QPR chairman remains committed to taking control at Valley Parade.
Paladini told the Telegraph & Argus: “All is good re Bradford. I am extremely serious about the purchase of the club.
“I am just waiting for a few more issues (to be resolved) but I don’t see a problem.
“I know we’ve gone over the deadline but all is okay and I hope to finalise the deal very shortly.”
Bradford could now be ready to listen to other potential investors who have been closely following events at Valley Parade.
Lawn has made it clear he and joint chairman Julian Rhodes cannot afford to take the club higher than the Sky Bet Championship.
When asked if he felt the Paladini deal was dead in the water, Lawn added: “Paladini did not fulfill the criteria of the exclusivity period, but not necessarily, no.
“You don’t know how it’s going to end.”