The House of Lords voted by 199 to 195 to approve an amendment allowing the referendum to be held any time before October 31 rather than fixing the date as May 5, the day of the Holyrood poll.
Lord Foulkes, who has warned holding the two votes on the same day is a recipe for confusion, did not press his own amendment that the referendum "must not" be held on May 5.
But he said there was now a big question mark over whether the referendum - on introducing the Alternative Vote system, where voters rank candidates in order of preference - could go ahead on the Government's favoured day.
He said: "We are scrutinising the Bill in very great detail. It's clear the committee stage will go into the New Year and it is increasingly less likely it will be technically possible to have the referendum on May 5."
The Government says two votes on one day will save money.