The former Leicester boss is in advanced talks with the Robins after a move materialised on Sunday, the PA news agency understands.
Pearson has been out of work since leaving Watford in July but is now expected to replace Dean Holden, after he was sacked by the Robins last week.
The Robins go to Middlesbrough on Tuesday with Pearson's appointment expected to be confirmed ahead of the trip to the Riverside.
Bristol City had also approached ex-Wigan boss Cook last week and enquired about MK Dons chief Russell Martin.
Cook was a contender to replace Lee Johnson last summer before the Robins appointed Holden.
MK Dons boss Martin has been impressing in his first managerial role after taking charge in November 2019.
Martin has the Dons 11th in Sky Bet League One having won seven of their last 10 games following Saturday's 4-3 victory over Northampton.
The PA news agency understands the Robins were put off by the £400,000 compensation they would have to pay and Martin remains happy at the club.
Bristol City are 15th in the Sky Bet Championship having lost their last seven games after Saturday's 1-0 home defeat to Barnsley.
They parted company with Holden last Tuesday after a 2-0 defeat to Reading.