The 20 year-old striker, who finished as Derry City's top goalscorer during his 12 month loan stint at Brandywell in 2019, is contracted to the Blades until the end of the current Premiership campaign.
However, it's understood the player, who had been training with the Sheffield United first team for the past number of weeks, believes it would be in his best interests to move on in search of regular first team football.
With Scottish champions, Celtic, Everton and Blackburn all reportedly interested in his services, Parkhouse could find himself with plenty of options ahead of a potential January Transfer Window move.
Sheffield United is understood to highly rate the youngster who was named in the PFAI Team of the of the Year 2019 and nominated for the Young Player of the Year award having scored 11 league goals and 19 in all competitions and had offered him a one year extension to his current deal.
However, Parkhouse is currently back home in Ireland assessing his options and hoping to secure a permanent move away from Bramall Lane after Christmas.