Mr Straw made clear he did not think the issue was going to go away and said there was "a sufficient case...for there to be further questions which will need to be dealt with".
His remarks came after the revelation that a dossier of 1000 emails had been handed to police purporting to show how Unite had plotted to undermine Labour's internal investigation into claims the union had sought to cram the Falkirk Labour Party with its supporters to ensure its preferred candidate won a local selection. Unite strenuously denies any wrongdoing. On Monday, Stevie Deans, a union official, resigned following an investigation by Ineos, owner of the Grangemouth petrochemical plant, into claims he used company time to work for Labour.
Mr Straw said the Falkirk row ran into the "catastrophic tactics" employed by Unite at Grangemouth. Last week, the union, having strongly opposed the management survival plan, performed a U-turn to accept it and in so doing helped save the jobs of 800 workers.
But the Labour grandee attacked Unite's strategy, saying: "Len McCluskey(the union General Secretary) downwards put internal Unite politics before the interests of their members at the Grangemouth plant."
Mr Straw stressed how he felt sure Mr Miliband would now look "very carefully" into whether the party's own inquiry should be reopened.
Meantime, Labour swept aside a claim by independent Falkirk MP Eric Joyce that it was keeping Mr Deans as chairman of Falkirk CLP out of fear of the Unite leadership. A senior Labour source, said the party HQ had no remit on the selection of local chairmen.