Ninth President of Israel Shimon Peres' office, together with the Sheba hospital, reports that Peres passed the night in stable condition. His status is still deemed serious, but his doctors have decided against operating on him at this stage.
The former President's family is being updated and consulted with throughout the treatment.
Hemi Peres, the former President's son, says he is optimistic about his father's condition improving.
"From time to time they're waking him up to check on his progress, and I've been with him at these times," Hemi told Galei Tzahal-Army Radio, "I have to say that at these times when he was conscious and I talked to him, I felt that he is getting better as time goes on.
Speaking outside the hospital near Tel Aviv where Peres is being treated, personal doctor and son-in-law Rafi Walden said the 93-year-old elder statesman was initially admitted to hospital on Tuesday as a precaution.
"Yesterday, Mr. Peres in his office delivered a remarkable speech for about an hour to a very enthusiastic audience," he said.
"Then he had vague ill feelings, and just to be on the safe side he was taken to hospital for examination. In the hospital, he suffered a stroke, with a relative loss of consciousness and other neurological symptoms."
A CT scan revealed a stroke on the right side of his brain. He was then intubated and sedated.
When his sedation level was lowered during the night, Peres was responsive, Walden said.
"When we lessened the sedation, he woke up - not completely but definitely was responsive to our appeals to him.
"He squeezed my hand and was definitely listening and understanding what was happening."
Walden said afterwards "we sedated him again just to keep him as quiet and peaceful as possible so as to enable the brain to recover."
He described Peres as "in a better condition than last night." Another attempt to lower his sedation level will be made in the afternoon, with an update expected around 4:00 pm (1300 GMT).
"There is no immediate threat to his life," Walden said.
"'[As for] the neurological condition, nobody can tell what will be the outcome but we are keeping optimistic ... I can't really say full recovery, but we are keeping optimistic and hoping for the best."
National and political figures have been expressing their hopes and prayers for Peres' full recovery ever since the news of his stroke hit. Prime Minister Binyamin Netanyahu wrote: "Shimon, we love you and all the people wish you a speedy recovery." Netanyahu has been getting continuous updates from the hospital.
President Reuven Rivlin said: "I am following with concern the updates from the hospital, and pray together with the entire people for my friend Shimon's recovery."
Chief Rabbis Yitzchak Yosef and David Lau also published statements wishing Peres a full recovery.
Rabbi Lau: "Peres has many merits in heaven from his decades of work on behalf of the nation of Israel, especially the strengthening of Israel's security since its establishment. It is fitting for all to join in prayer for his recovery at this time."
Sephardic Chief Rabbi Yitzchak Yosef also said everyone should pray for Peres' recovery: "He has the merit of saving the world of Torah study. He initiated the exemption from army service for yeshiva students, and he's always had great appreciation for Torah."