Last week, Kennedy wrote a letter to Massachusetts legislative leaders, asking that a 2004 law be modified to allow the governor to appoint a temporary successor before a special election is held.
Kennedy, who had been battling brain cancer for more than a year, died Tuesday at his home in Hyannis, Mass. He was 77.
In his letter, Kennedy wrote it was "vital" that Massachusetts have "two voices ... and two votes" in the Senate during the five months before a special election is conducted.
State Democrats were cool to Kennedy's plea while Republicans were against it, noting a similar request when Republican Gov. Mitt Romney led the state was opposed by Democrats.
State lawmakers were not scheduled to return to the Capitol until after Labor Day.
While Massachusetts voters would likely vote in another Democratic senator in the special election, delays could hamper efforts by Senate Democrats to round up the 60 votes needed to move healthcare legislation forward.