Senator Kitching, who died on March 10 aged 52 of a suspected heart attack, was farewelled by her family and MPs from across the political spectrum in Melbourne on Monday.
Mr Albanese says he will write to the ALP National Executive to ask for the establishment of a Kimberley Kitching Human Rights Award in recognition of her parliamentary work.
"This is a practical way to honour Kimberley Kitching's legacy and commitment to international human rights," Mr Albanese told The Daily Telegraph on Tuesday.
The award would be handed out at ALP National Conferences, which take place every three years, and will reportedly be the only award bestowed by the National Executive.
During her funeral on Monday, Senator Kitching's husband of more than 20 years, Andrew Landeryou, made reference to the "unpleasantness of a cantankerous cabal, not all of them in parliament, that was aimed at Kimba".
Following her death, reports emerged the senator had complained of being ostracised and bullied by members of Labor's Senate leadership team.
Mr Albanese has stopped short of setting up an inquiry into the allegations but did reiterate Labor would review its complaints processes.
"We will continue to examine all our internal processes and we will continue to listen to any constructive suggestions on how we can improve our processes," he told the Telegraph.
"We have a constant review of our internal processes and are always looking to improve them."
As the senator was farewelled at St Patrick's Cathedral her casket was adorned with an Australian flag and flowers, alongside a large floral "Kimba" sign.
Mr Albanese and a cast of federal Labor MPs, Deputy Prime Minister Barnaby Joyce, Defence Minister Peter Dutton, Attorney-General Michaelia Cash, Finance Minister Simon Birmingham and Victorian Premier Daniel Andrews all attended the service.
Also present was former Liberal prime minister Tony Abbott and secretary of the Victorian branch of the CFMEU John Setka.
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