According to Football Insider, the Blues are one of several clubs keen on the six-goal forward.
House signed for Eastleigh on a free transfer in August after being released by Reading.
He made just one appearance for the Royals and had loan spells with Swindon and Dagenham & Redbridge before his departure.
House has enjoyed a fine start for the Spitfires, who sit fifth in the National League table.
The 21-year-old’s scored six goals in eight appearances this season. His latest effort arrived in Eastleigh’s 3-1 win at Yeovil on Tuesday night.
House’s form in front of goal has reportedly led to Pompey’s attention.
But League One promotion rivals Ipswich Town and Charlton Athletic are also supposedly keeping tabs on the Scotland under-21 internatonal.
As things stand, Pompey have three senior striking options.
John Marquis is League One’s current top scorer with nine goals, while Ellis Harrison is another valuable player who provides something different.
Meanwhile, Jordy Hiwula was recruited on a free transfer in October. He penned a short-term deal until January 20, but has featured for only eight minutes in the league so far.