The website, which bore the logo of the Prime Minister's Office (PMO), misled some foreign news organisations into mistakenly reporting Lee's death last night.
The PMO said last night that it had lodged a police report about the website.
It added that no new information had been issued about Lee's health since an update earlier in the day that said Lee,91, was critically ill.
Police have confirmed that a report had been lodged and said they were looking into the matter.
"We take a very stern view against anyone who doctors a Government website to spread false information to deceive the public," Assistant Commissioner (AC) of Police Melvin Yong said.
"We will spare no effort to bring them to task. We also advise the public not to spread falsehoods," added AC Yong, who leads the police's Public Affairs Department.
A number of foreign media outlets carried erroneous reports about Lee's death but later withdrew the reports.
Lee's worsening health was first announced on Tuesday, when the PMO said he had an infection and was being treated with antibiotics.
It said doctors were monitoring his condition.
Lee has been in hospital since February 5 with severe pneumonia.
On February 28, the PMO said his condition had improved slightly, and he remained on antibiotics.
He is on a ventilator in the intensive care unit of Singapore General Hospital.
Born in 1923, Lee co-founded the city state in 1965 when it declared its independence from Malaysia.

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