"That is the democratic guarantee that now is there because of Labour Members of Parliament and 11 Conservatives joining with us."
Mr Gardiner said after Brexit the UK had to remain "closely aligned with our major trading partner" - the EU - which could mean continued membership of the single market.
He told Sky News' Sunday with Niall Paterson: "We haven't swept either the single market or a customs union off the table. We have said we are not fixated on the structures, what we want are the benefits."
Ms Abbott also told Marr that Britain’s businesses, the NHS and schools would be in a "terrible position" if the number of "eastern European migrants" stopped post Brexit.
She said that the Government should be listening to the concerns of the public sector and their fears of a "collapse" in the number of EU migrants moving to the UK.
She said: "The health service they're very worried about a collapse in the number of EU migrants coming here.
"Social care would be in a terrible position, the health service, finance, education, so we will be listening as the Government should be listening to what business and the public sector says about its needs for labour.
"At this point both business and public services like health and education are saying we do indeed need these eastern European migrants that are coming here.
"The reality is that business, the CBI, the Institute of Directors but also health, education and social care they say that they need these European migrants and we have to listen to them."
Asked to elaborate on shadow Brexit secretary Sir KeirStarmer's call for easy movement, Ms Abbot said Labour would put in place a "fair and reasonable" system to manage migration.
She added: "It will be relatively less bureaucratic than some of the proposals the Government have made."
She added: "We will have to see how this negotiation goes. It may involve a visa system, but we have to see how these negotiations the Government's undertaking go."
