Scottish Labour leader Richard Leonard is facing calls for the party to back permanent single market membership at its forthcoming conference.
Ten of Scotland's 73 Constituency Labour Parties (CLPs) have passed motions urging the conference to support the UK retaining permanent membership of the European single market and customs union.
Mr Leonard will face his first party conference as Scottish Labour leader in Dundee early in March.
It comes as UK leader Jeremy Corbyn is facing increased pressure to say Labour would commit to keeping the UK in the single market.
A letter backed by more than 80 senior Labour figures including Chuka Umunna and Lord Kinnock has warned that plans for investment in schools, hospitals and social care would be unfundable unless the UK stays in the single market.
The CLPs putting pressure on Mr Leonard are East Kilbride, Edinburgh Pentlands, Glasgow Anniesland, Inverness and Nairn, Western Isles, Paisley, Edinburgh Western, Greenock and Inverclyde, Glasgow Kelvin and Edinburgh Southern - the largest in Scotland.
Edinburgh South MP Ian Murray said: "Labour members in Edinburgh South and across the country are making their voices heard - they want our party to support permanent membership of the single market and the customs union.
"Given the strength of feeling among ordinary members, it's vital that we debate this at conference.
"If we are to leave the EU, the least-worst option for limiting the damage caused by Brexit is to remain as a participant in the single market and customs union. This is the only way to tackle austerity, protect jobs, and defend our hard-won rights for workers and consumers.
"Given the arithmetic in the Commons and the fact that a number of Conservative backbenchers support this stance, our continued participation in the single market and customs union is in Labour's hands.
"Our party was founded to protect the workers of the UK: if we fail them now we will never be forgiven."
A Scottish Labour party spokeswoman said: "Motions for debate are a matter for the conference arrangements committee."