It is believed a row between the two men escalated on Friday afternoon.
Merseyside Police said: “We have arrested a man following an incident at a construction site in Toxteth this afternoon.
“At 2pm, a man was reported to have been injured during an incident at a construction site at the junction of Stanhope Street and Plaza Boulevard.
“A 30-year-old man sustained a one-inch stab wound to his back and was treated at the scene by paramedics.
“He has been taken to hospital where his condition is described as stable. His injury is not believed to be life-threatening.
“A 37-year-old was arrested on suspicion of assault. He has been taken to hospital for treatment to a minor cut to his leg.
“A knife was recovered from the scene and taken for forensic testing.
“Further enquiries into the incident are ongoing. Witnesses are being spoken to and CCTV opportunities are being explored.”