The Britons, including a husband and wife from a Christian medical charity, were seized last Friday in Delta state, where they had been providing free health treatment.
Delta state police spokesman Andrew Animaka said there was still no update as to their whereabouts.
But he added: “We are following up on a lead with the arrest of four persons in connection with the incident.
“The suspects are currently in the custody of the state police command, in (the state capital) Asaba, and are rendering useful information.
“I can assure that we are on the heels of the abductors.”
Kidnapping for ransom is rife in southern Nigeria, where high-profile individuals and their families are a frequent targeted for criminal gangs.
Victims are usually released after a few days once payment is made.
Last Thursday, an Italian priest based in Nigeria for the last three years was kidnapped by armed gunmen near Benin City, the capital of Edo state.
He was released on Tuesday evening.
Father Maurizio Pallu told Vatican Radio from the Nigerian capital, Abuja, on Wednesday that it was the second time in a year that he had been kidnapped.
British diplomats in Nigeria have refused to comment about the latest abduction.
Police believe militants who have attacked oil and gas pipelines in the oil-rich Niger delta region are behind the kidnapping.
Nigerian intelligence sources said the Britons are thought to have been taken to militant camps in the creeks and swamps of the delta.