Deputy First Minister Nicola Sturgeon called for urgent talks with the commission last week after its president suggested a newly independent Scotland would need to apply for EU membership.
The comments by Jose Manual Barroso contradicted the SNP's long-standing claim that Scottish entry would be automatic after a Yes vote, and that Scotland would inherit the UK's opt-out from the euro and the open-border Schengen travel area.
The news that a meeting with Barroso is off the agenda for at least a couple of weeks is another setback after a week in which many within the SNP admit serious mistakes were made.
SNP MEP Alyn Smith said the party mishandled some of the issues around Europe and it was time for some "humility".
Referring to the related row over the SNP's lack of legal advice on EU entry, he told the Sunday Herald: "We took a sore one because of the legal advice that wasn't.
"That was just an unforgiveable own goal. We opened the door [to our opponents] with the use of the word 'automatic', which we should never have done. But we now have a realistic and robust position."