Lola Karimova-Tillyaeva posted a message of thanks on social media "for your kind words of support and best wishes for the speedy recovery of our president."
"It means the world to us, and I am sure that your heartfelt good wishes are helping in his recovery," wrote Karimova-Tillyaeva, Uzbekistan's ambassador to UNESCO.
The statement was the first on Karimov's health since his youngest daughter announced on Monday that he was in a "stable" condition in intensive care after suffering a cerebral haemorrhage over the weekend.
While information is very tightly controlled in the ex-Soviet nation, reports have appeared in opposition media based abroad claiming that Karimov is dead.
Anonymous sources in Uzbekistan denied the reports in remarks to Russian news agencies.
Uzbek officials have made no mention of Karimov's health since a terse government statement on Sunday saying he had been "hospitalised".
Doctors from Moscow's Burdenko neurosurgical hospital have been treating Karimov "from the very first moment," its head Alexander Konovalov told RBK news site today, while giving no details of his condition.
Tashkent authorities said fireworks planned for tomorrow's evening had been postponed indefinitely, saying the decision was due to the national team playing Syria in a qualifier match for the 2018 World Cup.
Karimov's press service today also released a written message from the president for Independence Day saying the country took the "correct path" in 1991.
He said Uzbekistan had transformed itself from a "backward" country with a one-sided economy focused on cotton to "a dynamically and stably developing powerful state."
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