Lord McConnell called for “fundamental change” in power structures, including a Nations and Regions Secretary of equal standing to a Deputy Prime Minister.
The Labour peer, who led Holyrood’s Labour and Liberal Democrat coalition between 2001 and 2007, also questioned the relevance of the Scottish Office.
Lord Wallace, deputy first minister for six years, backed plans for a UK Council of Ministers.
He said de-facto Deputy Prime Minister David Lidington “probably has more clout” on Scottish matters than Scottish Secretary David Mundell.
Giving evidence to Westminster’s Scottish Affairs Committee, Lord McConnell said: “I think after Brexit, when you have quite clearly some of the powers that are being repatriated from Brussels but also Britain’s new role in the world in terms of trade deals and so on, that I think so many of these areas involve both devolved and reserved powers that I think there is a need to have a new structure.”
He added: “I think there needs to be a fundamental change. I think in the UK Cabinet there should be a Secretary of State, either for the nations and regions or for constitutional affairs, that person in my view would almost automatically be a sort of Deputy Prime Minister-type individual.”
Lord McConnell said there would no longer be a Scottish Secretary, adding following devolution in 1999 he was “surprised ... that there was still a Secretary of State for Scotland”.
He added: “Alongside that ... you need a formal process for decision-making on areas where there is clearly going to be shared legislative responsibility post-Brexit.
“That’s why alongside that department I think you should have a UK Council of Ministers.”
He said it would allow ministers for each of the devolved powers and the UK to meet on an “equal basis” and make joint decisions for the UK in shared areas, giving “proper respect and weight” to all involved on areas such as fishing and agriculture.
Questioned if his proposal would give the Scottish Government a veto over UK Government decisions, he suggested majority voting could apply to decisions of the council but that would “seriously threaten the sensibilities of the UK Government”.
Lord McConnell also called for the House of Lords to be replaced with a House of the Nations and Regions.
Giving evidence to the committee, Lord Wallace said: “I do agree that there should be a UK Council of Ministers.”
He added: “I think as we move forward there will be areas, agriculture is an obvious one, where there is logic in having an overall, overarching framework and that the respective administrations then implement it for their geographical area.”
He said there would be an issue regarding England not having a devolved government but he does not see any appetite for English devolution.
The Liberal Democrat peer also questioned the power of the Scotland Office.
“I suspect [David Lidington] carries more clout for Scotland’s interests than Mr Mundell does, no disrespect to Mr Mundell,” he said.
“I don’t think we would be have been any worse off in the early days of devolution if the Scotland Office had disappeared.”