The PM was taken to a London hospital on Sunday evening with "persistent symptoms" - including a temperature - for a series of routine tests.
It is said to be a "precautionary step" taken on the advice of his doctor.
Mr Johnson, 55, tested positive for coronavirus 10 days ago.
Foreign Secretary Dominic Raab will chair Monday morning's coronavirus meeting in his place, Mr Jenrick said.
Last month, the prime minister's spokesman said if Mr Johnson was unwell and unable to work, Mr Raab, as the first secretary of state, would stand in.
Mr Jenrick told BBC Breakfast: "We hope that as a result of these tests [the prime minister] will be able to come back to Downing Street as soon as possible.
"He's been working extremely hard leading the government and being constantly updated. That's going to continue."
"I'm sure this is very frustrating for him, for somebody like Boris who wants to be hands [on] running the government from the front, but nonetheless he's still very much in charge of the government," he added.
US President Donald Trump is among those who has sent his wishes to Mr Johnson.
"All Americans are praying for him. He's a great friend of mine, a great gentleman and a great leader," Mr Trump said, adding that he was sure the prime minister would be fine because he is "a strong person".
And Labour leader Keir Starmer said he hoped the prime minister had a "speedy recovery".
Health Minister Nadine Dorries, who herself tested positive for coronavirus last month, said many of those with the virus would be "felled" by fatigue and a high temperature and use isolation to sleep and recover.
"Boris has risked his health and worked every day on our behalf to lead the battle against this vile virus," she said in a tweet.
Presentational grey line

Coronavirus is straining the highest levels of government

Analysis box by Chris Mason, political correspondent
The prime minister, alongside the Queen, personifies the country's public response to this pandemic.
And Boris Johnson is continuing to personally experience the unpleasant reality of the virus.
Downing Street officials are adamant Mr Johnson remains in charge of the government and is in contact with ministerial colleagues and civil servants.
But the undeniable reality is there is nothing conventional, nothing normal about this - however routine the tests are that the prime minister is receiving.
The coronavirus has repeatedly proven its capacity to turn the far-fetched into reality, over and over again.
Advisers, officials and ministerial colleagues have all been forced to self-isolate.
Covid-19, the illness which the virus causes, is crippling the economy, robbing us of our usual liberties - and now it is straining the personal capacity of those at the highest level of government to respond to it.
Presentational grey line
Dr Sarah Jarvis, a GP and broadcaster, told the BBC that Mr Johnson would be likely to have his chest X-rayed and his lungs scanned, particularly if he had been struggling for breath.
She said he is also likely to have an electrocardiogram to check his heart's function, as well as tests on his oxygen levels, white blood cell count, and liver and kidney function before he is released from hospital.
Mr Johnson has worked from home since it was announced that he had tested positive for coronavirus on 27 March.
He was last seen in public applauding the NHS and other key workers from his flat in Downing Street on Thursday evening, and chaired a coronavirus meeting remotely on Friday morning.
Also on Friday, the prime minister posted a Twitter video in which said he was still displaying minor. So in accordance with government advice I must continue my self isolation until that symptom itself goes," he said.
"But we're working clearly the whole time on our programme to beat the virus."
On Saturday, his pregnant partner Carrie Symonds tweeted that she had spent a week in bed with the main symptoms.
She said she had not been tested for the virus.
Health Secretary Matt Hancock had also tested positive for the virus and returned from self-isolation on Thursday to host the daily Downing Street news conference.
The government's chief medical adviser, Prof Chris Whitty, has also had to self-isolate after showing symptoms.
The news of Mr Johnson's admission to hospital came shortly after the Queen delivered a rallying message to the nation, saying the UK "will succeed" in its fight against the coronavirus pandemic.
In a rare speech, the monarch thanked people for following government rules to stay at home and praised those "coming together to help others".