Nicola Sturgeon and Mark Drakeford believe the EU Withdrawal Agreement proposals are being rushed through and said it is "imperative" they are subject to "detailed scrutiny".
They are urging MPs and the UK Government to secure an extension to the Article 50 process.
Mr Johnson has now been forced to put his plans to leave the EU on October 31 on hold after MPs voted by 322 to 308 to reject his bid to push legislation approving his Brexit deal through the Commons in just three days.
Scotland's First Minister Nicola Sturgeon and Mr Drakeford, her Welsh counterpart, will hold a joint press conference on Wednesday in London where she is expected to say: "The UK Government has sought to pass the Withdrawal Bill through Westminster without an opportunity for detailed scrutiny in either the UK or Scottish Parliaments. That is unacceptable.
"It is imperative that this Bill is subject to detailed scrutiny in all of the UK's legislatures and that the views of the Scottish Parliament and Welsh Assembly are taken into account.
"Any form of Brexit will damage our economy, which is why the Scottish Government has been clear and consistent that the best option for the future wellbeing and prosperity of Scotland is to stay in the European Union."
The Prime Minister on Tuesday told MPs he would now "pause" the Withdrawal Agreement Bill until the EU takes a decision on whether to grant another Brexit delay.
He said: "I will speak to EU member states about their intentions. Until they have reached a decision we will pause this legislation."
He added: "Let me be clear. Our policy remains that we should not delay, that we should leave the EU on October 31 and that is what I will say to the EU and I will report back to the House.
"And one way or another we will leave the EU with this deal, to which this House has just given its assent."
The two First Ministers have jointly written to Mr Johnson and the President of the European Council asking for an extension to give more time to scrutinise the Bill.
At their press conference, they will also say that Brexit would damage devolution and be disastrous for the economies of both Scotland and Wales.
Mr Drakeford will say: "This deal is a bad one for both Wales and Scotland.
"Yesterday the Senedd agreed that we could not give legislative consent to the Withdrawal Bill as currently drafted.
"The First Minister of Scotland and I have jointly written to the Prime Minister and the President of the European Council asking for an extension so that the Scottish Parliament and the Senedd can fulfil their constitutional duty and properly scrutinise the Withdrawal Agreement Bill.
"There is another way, a referendum which gives people the chance to accept Johnson's deal or choose to remain in the EU. We think remain is the right answer for Wales."