Not really, at least not that we know about it. Knowledge of the Senate in this period is rather fragmentary and it seemed to simply just fade away. The barbarian overtake of the Italy marked short period of the rise of the importance of the Senate, as Germanic kings often use it to give legitimacy to they actions (Odoacer's use of the Senate in "ending" Roman Empire is good example of that).

During Ostrogoth-Byzantine war, the Senate (or at least number of prominent senators) thrown they lot with Justinian I and response from king Teia in 552 cripled Senate beyond repair.

The institution itself dragged on for at least half a century beyond that, but it's importance did not extended outside Rome. Byzantine prefect Longinus in 568 noted the Roman Senate disappeared gradually, however it might be anachronism (as this note comes from the 9th century chronicle) as the last mentioning of the Senate comes from 603, but it also might be indication how insignificant Senate become.

N.B. Upon dividing the Empire, Constantine I created separate Senate for the East- that one lasted until the Fourth Crusade.

As for office of consul specifically, at this time period they were appointed by emperor, as any other position with any and all elections being only a rubber stamp for traditions sake.

Consuls were dutifully "elected" for both East and West until reforms of Justinian I, who abolished office entirely. (Technically, it was done by Leo VI, but Justinian merged it with the title of the emperor so effectively quite the same thing).