The 70,000 square feet of Grade A office development is located in the Monks Cross area.
CBRE and Carter Towler are properties agents.
The MCR Property Group refused to disclose the amount paid to the property.
Moorside tenants include Aviva, Garbutt & Elliott, Rollits Solicitors, Portakabin and Shepherd Group.
MCR fund director Martin Brown commented: “This is a significant acquisition of MCR that will strengthen its position in this region of the country, not just in the office sector but also in other active projects in Bridlington and surrounding areas.
“York is a key area in the region and is considered a commercial hub for Yorkshire, attracting businesses keen to make this thriving city their headquarters.
“Our aim is to integrate Moorside's position as a key commercial position between now and the future.
“While this site has ambitious plans, our immediate focus is to ensure that the transition is positive for all parties.”