The former Scottish Labour leader has campaigned strongly for a second referendum in the hope of reversing the Brexit decision and says she wants to see pro-Remain candidates if the elections go ahead.
She would not comment on the idea, but one source said she was considering standing.
“She feels so strongly about Scotland’s links with Europe it would make every sense – and if it’s a short spell she would not have to give up as an MSP,” the source said.
Prime Minister Theresa May wants to avoid the UK taking part in the European elections taking place next month, but the EU has said that will be a condition of a further extension of the Brexit deadline.
Labour won two of the six Scottish seats in the European Parliament at the last elections in 2014, with two going to the SNP, one to the Tories and one to Ukip.
Labour’s Catherine Stihler has already stepped down after finding another job and David Martin, an MEP since 1984, has yet to decide whether he would stand again.
Ms Dugdale said: “It has not even been confirmed there will be European elections yet. If there are, I would be very keen to see pro-Remain candidates standing.”