British Foreign Secretary Dominic Raab said he was "deeply concerned" by the reports Mr Navalny had been poisoned, and sent his thoughts to him and his family.
What has the spokeswoman said?
Kira Yarmysh, the press secretary for the Anti-Corruption Foundation, which Mr Navalny founded in 2011, tweeted: "This morning Navalny was returning to Moscow from Tomsk.
"During the flight, he felt ill. The plane made an urgent landing in Omsk. Alexei has toxic poisoning."
She added: "We suspect that Alexei was poisoned by something mixed into [his] tea. It was the only thing he drank since morning.
"Doctors are saying that the toxic agent absorbed faster through the hot liquid. Right now Alexei is unconscious."
Ms Yarmysh later tweeted that Mr Navalny was on a ventilator and in a coma, and that the hospital was now full of police officers. She said they had later asked to search his belongings.
She also said that doctors were initially ready to share any information but then they later claimed the toxicology tests had been delayed and were "clearly playing for time, and not saying what they know".
She said in a tweet at 14:58 local time (08:58 GMT) that Mr Navalny's condition had not changed and he was still unconscious.
She was told the diagnosis would be "towards evening".
Both Mr Navalny's wife, Yulia Navalnaya and doctor, Anastasia Vasiliyeva, had arrived at the hospital but were being denied access to him, Ms Yarmysh said.
What are the other reports from the scene?
The Tass news agency quoted one source at the Omsk Emergency Hospital as saying: "Alexei Anatolyevich Navalny, born in 1976. Poisoning intensive care."
However, the deputy head physician of the hospital later told media that it was not certain Mr Navalny had been poisoned, although poisoning was "naturally" one of the diagnoses being considered.
Anatoly Kalinichenko said that doctors were "genuinely trying to save [Mr Navalny's] life".
Kremlin spokesman Dmitry Peskov later said it wished the critic a speedy recovery - as it would all citizens in such circumstances - and that the authorities would consider approving treatment abroad if it were requested.
Video footage on social media shows Mr Navalny being taken on a stretcher to an ambulance on the airport runway.
Other disturbing video appears to show a stricken Mr Navalny in pain on the flight.
Passenger Pavel Lebedev said: "At the start of the flight he went to the toilet and didn't come back. He started feeling really sick. They struggled to bring him round and he was screaming in pain."
Another photograph on social media purports to show Mr Navalny drinking from a cup at a Tomsk airport cafe.
The Interfax agency said the cafe owners were checking CCTV to see if it could provide any evidence.
Who is Alexei Navalny?
He made a name for himself by exposing official corruption, labelling Mr Putin's United Russia as "the party of crooks and thieves", and has served several jail terms.