Defender Burgess, 22, joined Pools at the start of August and Hartlepool-born midfielder Dolan, 20, arrived later the same month.
Both players joined on loan until January and have been regular starters for Cooper's side this season.
"I had a productive discussion with the Middlesbrough chief executive and as a result I'm hopeful that Christian and Matty can stay with us," Cooper told Pools' official website.
"We'll work more on it during the next couple of days and hopefully come up with an agreement because we'd love them both to stay with us for the rest of the season if possible."
Burgess is scheduled to play his last game for the club at Morecambe on Sunday and Dolan is due to return to the Riverside Stadium following the home game against Bury on New Year's Day.