"The problem is most of his Labour MPs have more allegiance to Tony Blair than they do to Jeremy Corbyn so we now have a hopelessly divided Labour Party", Sidney Cordle, the party leader, said.
Ms Eagle, who has been tipped as a contender in a leadership contest, said: "The country doesn't have an effective opposition at the moment".
Party sources say that among those who filled the "why I joined" box on the membership application form, at least 80% have said they did so to support Corbyn.
Mr Free added that the membership of SW Wilts Labour party had risen from 114 members pre-Corbyn to well over 600, including around 100 supporters, and has risen every week since Mr Corbyn was elected.
The assessment of the situation was echoed by a spokesman for Mr Corbyn following the talks between the Labour leader and the Unite chief.
The Unison forum backed a motion agreeing to continue to "support Jeremy Corbyn with his anti-austerity platform and his belief in a new kind of politics that is progressive and fair in its outlook".
He said Mr Corbyn would need to secure backing from more than 50 MPs if he wanted to fight a leadership challenge.
The 61-year-old father-of-three, who has represented his seat since 1997, abstained in last week's vote of no confidence in Mr Corbyn and told members of his constituency party that he thought the vote's result was unconstitutional.
Chants of "Corbyn in, Tory's out" echoed round New Street as the demonstration, organised by members of pro-Corbyn Labour campaign group Momentum, made its way through the city centre.
"I have a huge responsibility", Corbyn said.
Tom Watson told MPs: "It is the last throw of the dice".
Attitudes are hardening on both sides the dispute to such an extent that it is likely when the executive makes its ruling, one side or the other will take out an injunction, so that a High Court judge will have to adjudicate on the exact meaning of the Labour Party rule book. "I don't think they will regard any settlement that sees Jeremy remain in place as being acceptable".
Vice chair of the Wallasey CLP Paul Davies said the meeting of the Wallasey branch saw 21 party members take part and 19 voted to support Jeremy Corbyn.
"The rules of the party are there, and if people want to have a contest, they should use those rules to trigger a contest - then we'll see".