13 June 2016 Last updated at 08:43 BST
Labour's Mr Hunt, for Remain, told the West Midlands EU referendum debate in Birmingham that in order to trade with the EU, the UK would "have to accept the free movement of labour".
However, Lord Jones said no-one knew the conditions of any future trade deals.
The Leave campaigner, who served as a trade minister in Gordon Brown's Labour government, told the MP: "There's a very good BBC word for this; it's called disingenuous, which means you're lying."
Stoke MP Tristram Hunt demanded an apology from Lord Digby Jones after being accused of lying about immigration during an EU debate.