The suggestion was made by Lord Adonis, who heads the pro-EU European Movement UK. Lord Adonis previously served under the leadership of Tony Blair and Gordon Brown.
Brussels is insisting Britain fully implement customs checks on goods travelling between Northern Ireland and the rest of the UK.
However, this is infuriating unionists who argue it undermines British sovereignty.
French President Emmanuel Macron has threatened to veto any compromise with the UK.
Taking to Twitter Lord Adonis said: “Boris Johnson’s Brexit deal has no friends.
“It’s over, Blair isn’t coming back either nor is PR. For your own sanity, support a project that may work.”
Lord Adonis’s message received over 3,000 likes and 600 retweets.
In a blow to re-joiner hopes, social media users replied to ask if the EU would want Britain back in its union.
Ann Gymer wrote: “Do you really think the EU would have us back?
“I’m not sure they will and I am saddened by that.”
Cheryl Russell added: “Do you think they’d let us rejoin? Really?”
Boris Johnson’s Brexit trade deal introduced a customs border down the Irish Sea for some products.
This was done to avoid a border between Northern Ireland and the Irish Republic.
However, unionists and loyalists are mounting a campaign of opposition, with hundreds protesting in Belfast’s Shankill Road on Thursday.
Mr Johnson is urging the EU to show greater flexibility to make the Protocol work.
Crunch Brexit talks will take place at the G7 on Saturday between Mr Johnson and European leaders.