Speaking to the Commons health committee on Tuesday afternoon, Vallance said it was “reasonable” to assume there were as many as 55,000 cases already.
And he told MPs the social distancing measures put in place by the government last night would have to last at least “months”.
The number of people to test positive for coronavirus in the UK as of 9am on Tuesday was 1,950, up from 1,543 on Monday, the Department of Health and Social Care said.
At least 56 patients who tested positive for coronavirus have died.
A total of 50,442 people have been tested with 48,492 negative results.
Vallance told MPs this afternoon the “hope” was the number of deaths could be kept down as a result of the government’s decision to try and stop being gathering in large numbers.
Boris Johnson last night dramatically called on people to stay away from pubs, clubs and theatres and to avoid all non-essential contacts and travel.
People who live with somebody who is ill have been told to stay home for 14 days.
Vallance said: “Every year in seasonal flu the number of deaths is thought to be 8,000.
“If we can get this down to numbers 20,000 and below, that is a good outcome in terms of where we would hope to get to with this outbreak. It is still horrible. That is still an enormous number of deaths.”
He said case isolation could bring the peak of the outbreak down by about 20%, whole household quarantine by about 25%, and general social distancing by about 50%, while social “shielding” of the elderly could reduce the mortality rate by 20% to 30%.
“Together you should expect those to have a very significant effect on the peak and we should start to see the rates come down in two or three weeks’ time,” he told the committee.
Jeremy Hunt, the former Conservative health secretary who now chairs the committee, asked Vallance whether there was “potentially 55,000 cases” in the UK at the moment.
Vallance said: “If you put all the modelling information together, that’s a reasonable ballpark way of looking at it.” He added: “It’s not more accurate than that.”
Asked if the social isolation and distancing measures could have to be kept in place for as long as 18 months, Vallance said: “I don’t know how long these measures are going to be needed for.
“Certainly not a couple of weeks. It’s going to be months. I don’t know how many months.”
Vallance also defended the government’s approach to tackling the virus. he said it was a “semantic difference” to argue whether the UK had shifted from a process of delaying and mitigating the spread of Covid-19 to an attempt to suppress it.
He told MPs the approach had always been to “save lives and protect the vulnerable” by delaying and suppressing the peak of the outbreak and shielding those most likely to be badly hit.
It came as NHS England announced it was suspending non-urgent elective surgery to free up beds for coronavirus patients.
Chief executive Sir Simon Stevens told MPs operations would be suspended from April 15 at the latest for at least three months.