Former president Nelson Mandela in 2008.
"Madiba remains in hospital in Pretoria, but his doctors have confirmed that his health is steadily improving," spokesperson Mac Maharaj said in a statement.
Mandela was admitted into a Pretoria hospital on June 8 with a recurrent lung infection.
Until now, his condition has been described as critical but stable.
Zuma said: "On behalf of government and all the people of South Africa, we wish Madiba a joyous 95th birthday … We are proud to call this international icon our own as South Africans and wish him good health."
Zindzi Mandela said her father was making "remarkable" progress in hospital.
"He responds very well … with his eyes, and he nods and sometimes he lifts his hand like to shake your hand," she told Britain's Sky News on Wednesday.
'He continues to amaze us'
"There was a time that we were all extremely anxious and worried, and we were prepared for the worst," said Zindzi. "But he continues to amaze us every day."
"We thank all our people for supporting Madiba throughout the hospitalisation with undying love and compassion," said President Jacob Zuma on Thursday.
"We also thank all for responding to the call to give Madiba the biggest birthday celebration ever this year."
Last week, former president Thabo Mbeki also suggested Mandela might be discharged from hospital soon.
Meanwhile, US President Barack Obama paid warm tribute to Mandela for his birthday.
Obama's message came as admirers around the world prepared to honour the former South African president's legacy with a wave of charitable acts.
Obama, who visited South Africa last month but was unable to see Mandela because of his illness, was fulsome in his praise of the former statesman.
"Our family was deeply moved by our visit to Madiba's former cell on Robben Island during our recent trip to South Africa, and we will forever draw strength and inspiration from his extraordinary example of moral courage, kindness, and humility," Obama said in a statement.
Mandela Day
In 2009, July 18 was declared as Mandela Day to honour the former statesman, with people encouraged to dedicate 67 minutes of their time to nation-building and charity in recognition of his service towards securing democracy and restoring human dignity.
Obama called on people to honour the elderly leader by taking part in Nelson Mandela Day on Thursday, when admirers around the world will uphold his legacy with charitable acts.
Biker gangs will clean the streets, volunteers will paint schools and politicians will spend 67 minutes on worthy projects – all to mark Mandela's 67 years of public service.
In Danville, Pretoria, Zuma himself will try to channel Mandela's cross-community appeal by delivering government housing to poor whites.
Children in schools around the country will kick off the day by singing "Happy Birthday" to the former prisoner, who also marks 15th anniversary of his marriage to third wife Graça Machel.
The government will also host a ceremony for the symbolic handing over of Mandela's new high-tech ID card, which will be received by Zindzi.
The event is laden with symbolism in a country where apartheid was enforced by pass books, which black citizens were forced to carry and which limited movement to certain areas at certain times.
'Make the world a better place'
Global luminaries, pop stars and ordinary people around the world have joined South Africans in pledging support for Mandela on his birthday.
"I will also be giving my 67 minutes to make the world a better place, one small step at a time," UK business magnate Richard Branson vowed in a recorded message.
In Manila, capital of the Philippines, 50 abandoned street children will get a television studio tour and see performances by local artists.
On Saturday, the Australian city of Melbourne will hold a concert featuring local and African artists, while a music festival later this year in Norway will promote equality in schools.
Negotiating the end to apartheid
Born on July 18,1918, Mandela fought against minority rule in South Africa as a young lawyer and was convicted of treason in 1964.
He spent the next 27 years in jail.
It was in part through his willingness to forgive his white jailers that Mandela made his indelible mark on history.
After negotiating an end to apartheid, he became South Africa's first black president, drawing a line under centuries of colonial and racist suppression.
He then led reconciliation in the deeply divided country.
Mandela's peace-making spirit has won him worldwide respect.
"Never before in history was one human being so universally acknowledged in his lifetime as the embodiment of magnanimity and reconciliation as Nelson Mandela," said Archbishop Desmond Tutu, himself a Nobel Peace laureate.
But the sunset of Mandela's life has been somewhat eclipsed by bitter infighting among his relatives.
A row over his final resting place has seen three of his children's graves dug up and their remains moved amid public brawling and legal action among his children and grandchildren. – Sapa, Reuters