He was 99.
"He had been admitted to a private hospital in the city for the past few days and was discharged last night," the member said.
Vaswani died in the mission premises this morning, she said.
The mission was planning a grand celebration on his 100th birthday next month, she said.
His mortal remains have been kept at the mission to enable people to pay their last respects.
Veteran BJP leader Lal Krishna Advani, a frequent visitor of the mission, is expected to attend Vaswani's last rites, she said.
"Sad to hear of the passing of Dada J P Vaswani, a spiritual leader who contributed so much to our society. He dedicated his life to the betterment of humanity, promoting simple living, high thinking and education. My condolences to his countless followers," the mission said in a release quoting the president.
Born on August 2, 1918 in Pakistan's Hyderabad city in a Sindhi family, Dada Vaswani, as he was popularly known, was the head of the Pune-based Sadhu Vaswani Mission, a non-profit organisation involved in social work and charity.
The organisation runs educational institutions and hospitals in the city.
Prime Minister Narendra Modi had addressed the spiritual leader's 99th birthday celebrations last year through a video link. Modi had then said Vaswani's blessings would help in accomplishing goals for a new India.
Acclaimed as a humanitarian, philosopher, educator, orator and non-sectarian spiritual leader, Dada Vaswani had captivated the hearts of millions of people worldwide. He was also into promoting vegetarianism.
His regular discourses at the mission premises attracted people from different walks of life.
Vaswani was a brilliant student, but he later gave up a career in academics to follow in the footsteps of his illustrious uncle and guru, Sadhu T L Vaswani, who had set up the charity organisation.
He was awarded a fellowship at the D J Sindh College (in Pakistan) for standing first in his class.
His MSc thesis on 'The scattering of X-Rays by Solids' was examined by Nobel laureate C V Raman, who was impressed by the originality of Vaswani's views.
As the news of Vaswani's death spread, people from all walks of life, including those who shared a good rapport with him, expressed their grief over his demise.
"Dada often would invite me for reactions to his views and projects. Some of my write-ups were published in the mission's publication," he said.
Dabre said when Vatican celebrated the golden jubilee of its project of inter-religious dialogue, he celebrated that at the Sadhu Vaswani Mission.
Imam Umer Ahmed Ilyasi, chief Imam of All India Imam Organisation, said Vaswani was a great soul, who always gave importance to humanity and spread love and harmony.
"We all should follow his footsteps and teachings," he said.
Lok Sabha member from Pune, Anil Shirole said, "Today we have lost a saint, who worked relentlessly towards spreading humanity and peace. The real homage to him will be to follow in his footsteps and teachings".
Vaswani has penned over 50 books in English and many more in the Sindhi language.
Many of his books have run into several editions and translated into Marathi, Hindi, Gujarati, Kannada, Papiomento (Portuguese-based creole language), Arabic, Mandarin, Spanish, French, German and Indonesian languages.
He had spoken at various prestigious international platforms and received the U Thant Peace Award in April 1998, for his dedicated service to the cause of world peace.
The Sadhu Vaswani Mission, having followers from all over the world with its centres in multiple nations, works in the fields of education, healthcare, social service, animal welfare and spirituality.