MORE: We've come so far - now for the big push He said: "Nearly two and a half years on from the referendum, we are, both on the European Union deal, and on other post Brexit trade deals, still lost in campaign mode on fantasy island".
Cabinet Office Minister David Lidington is also due to hold Brexit meetings with figures from the UK's devolved administrations on Thursday.
The EU proposed a "backstop" solution to keep the Northern Ireland in the bloc's customs union and parts of the single market that underpins the 1998 peace agreement there - an option British prime minister Theresa May has rejected so far arguing it would create a border within the United Kingdom, in the North Sea between Northern Ireland and the mainland UK.
The de-facto Deputy Prime Minister was pressed on the DUP's support when he appeared on ITV's Peston on Wednesday night.
He also stressed the EU's insistence that Britain must accept possible checks on goods moving between its mainland and its province of Northern Ireland, saying Brexit will trigger the need for customs, value-added tax and compliance checks with European Union standards.
But he said that checks on animals and animal food products passing between Northern Ireland and Great Britain would have to increase tenfold.
London hopes to resolve the issue with a future trade deal, but agrees there should be a "backstop" arrangement to avoid physical frontier checks until that deal is done.
The plan would also involve keeping Northern Ireland in the single market to help maintain frictionless trade across the border with the Republic while mainland Britain would be outwith the single market.
He suggested all 10 DUP MPs could vote down this month's budget to "pull the government back into keeping its promises".
Speaking to Sky News in his constituency of Larne on Northern Ireland's east coast, Mr Wilson said that the DUP were not going to be "bullied" or "bribed" into supporting a deal that could see Northern Ireland effectively kept in the single market or customs union.
"But that's not a question for us, we're not members of the Conservative Party".
The Prime Minister relies on the support of 10 of Northern Ireland's DUP MPs to prop her up in the House of Commons and to secure their backing senior Tories are ready to offer them a bribe.