Although it was not up to Britain alone to decide the future of the treaty, it would not be "sensible" for the EU Constitution -- which provides for a British vote -- to continue through Parliament, he said.
Straw refused to declare the constitution dead, telling Parliament the situation would be kept under review. The government reserved the right to reinstate the bill at any time, he added.
However, Conservative Shadow Foreign Secretary Liam Fox said the constitution should either be declared dead or the British people should be given a chance to lay it to rest once and for all.
Straw denied any parts of the treaty would be introduced "by the back door" but said certain changes to the workings of the European Union would have to be introduced for it to function effectively with 25 members.
Europe's Council of Ministers will discuss the treaty's future on June 16. Straw said he would detail the position Britain would take at that meeting next week.
Notable deaths of 2014 [PHOTOS]