Tagore had lived at No 3, Heath Villas in Hampstead Heath, north London, for a few months in 1912 while he translated his collection of poems 'Gitanjali'.
During a meeting with acting Indian High Commissioner to the UK, Dinesh Patnaik, on arriving in London for a week-long tour of the UK on yesterday, Banerjee expressed her state's desire to buy the privately-owned lodgings.
"The home has great historical significance and the chief minister is keen that it be turned into a memorial to Tagore," a person close to the discussions said.
She has revived her request this time with the hope of some movement on the plans.
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The home already has a blue plaque commemorating its famous former Indian resident.
"Rabindranath Tagore (1861-1941) Indian poet stayed here in 1912," reads the inscription.
Tagore had set sail for England from India in 1912 and was known to have translated many of his works while in London.
His company at the time included famous British artists and poets, including W B Yeats who also wrote the introduction to 'Gitanjali' the collection of 103 translations which went on to win Tagore his Nobel Prize for literature the next year in 1913.
A bronze statue of Tagore, commissioned by Tagore Centre UK andunveiled by Prince Charles in 2011, stands at Gordon Square in central London.
It now remains to be seen if the owners of the poet's London base in 1912 would be open to selling the property.
While in the UK, Banerjee is also scheduled to formally unveil a commemorative blue plaque in the memory of Sister Nivedita, the Scottish-Irish social worker and follower of Swami Vivekananda best known for her charitable work in Kolkata.