A draft agreement, uncovered by the Sunday Times, says that even where there is "overwhelming public interest" in speaking to journalists, the government must be told first.
The government said the wording was "standard".
It set up a £1.6bn fund last year to repair the most dangerous buildings.
But it warned that the fund might not cover all the costs of removing the cladding.
Some types of the covering, often added to newer blocks of flats, have been proven to be a fire hazard.
After the 2017 Grenfell fire, the government pledged that safe alternatives to dangerous cladding would be provided on all buildings in England taller than 18m.
It set up the £1.6bn fund to help foot the costs.
The agreement, between the building owner or leaseholder and the government, says: "The Applicant shall not make any communication to the press or any journalist or broadcaster regarding the Project or the Agreement (or the performance of it by any Party) without the prior written approval of Homes England and [the Ministry for Housing, Communities and Local Government ]" and its press offices.
It says an exception can be made "where such disclosure is in the overwhelming public interest (in which case disclosure will not be made without first allowing Homes England and MHCLG to make representations on such proposed disclosure)."
The UK Cladding Action Group tweeted that it was "clearly a matter of public interest" that these issues were aired in public.
"No department should be hiding behind non-disclosure agreements to stop scrutiny of their actions," the group said.
Another campaign group, Manchester Cladiators, said the existence of the "gagging clause" was "shocking but not necessarily that surprising".
Spokesperson Rebecca Fairclough said residents would feel "intimidated" by it, adding: "We ask the government to remove this unfair clause immediately and focus on the priority of solving this institutional failure, which still exists and is only growing over three and a half years after the Grenfell tragedy."
The government insists that the wording in the agreement, under the heading "Marketing material", is there to ensure applicants come to the government first.
"The terms set out are standard in commercial agreements and are not specific to this fund - to suggest otherwise is misleading and inaccurate," the Ministry for Housing, Communities and Local Government (MHCLG) said in a statement.
"We want a constructive working relationship with building owners who apply to the fund and applicants are asked to work with the department on public communications relating to the project."
I’d rather live with a good question than a bad answer.