A representative for Greenvale told Grower exclusively that the allegations "came to light when it was noticed by our accountants that various sums of money were moving in an inexplicable way".
He added: "They started an investigation and immediately informed Sainsbury's and the police and an investigation is [now] being carried out by the City of London Police Fraud Squad."
Reports in the national newspapers suggested that the alleged payments between the two men totalled some £3m.
City of London Police did not confirm this amount but said in a statement that two men were arrested on 13 March on suspicion of corruption and money laundering. They did not reveal the identity of the men, who were released on bail.
Greenvale, one of the country's main potato producers with three state-of-the-art packing plants in Shropshire, Berwickshire and Cambridgeshire, alerted police to the case following an internal investigation by its holding company Produce Investments. Greenvale said in a statement: "This matter was brought to the attention of J Sainsbury... following an internal investigation into payments made to individuals outside the group."
Greenvale, which supplies Sainsbury's with the majority of its potatoes, also told Grower that a number of staff members have been suspended and that the investigation is ongoing. "We have not finished the investigation... We do not know if there is more to come or not."
Sainsbury's said in a statement: "We can confirm that allegations have been made concerning certain payments and benefits by a supplier to an individual employed by us.
"We have the highest standards of corporate governance and following inquiries have passed our findings to the police for further investigation.
"As a result, we are unable to comment further at this stage."
A police representative said: "City of London police executed and completed search warrants at two residential addresses in Cambridgeshire and Shropshire and two business premises in Shropshire on Thursday 13 March. Two men were arrested on suspicion of corruption and money laundering and were subsequently bailed."