Wimbledon management have promised to "act responsibly" and insist they are preparing for the tournament to go ahead on schedule.
The French Open was moved on Tuesday from a May start date to September, taking players by surprise, with suggestions the men's and women's tours may also have been caught out.
In the fast-moving climate of concern over the coronavirus pandemic, the All England Lawn Tennis & Croquet Club (AELTC) is preparing for Wimbledon to begin on June 29, but there is acknowledgement that may not be possible.
The AELTC said on Tuesday it has closed down parts of its grounds, including its museum, and many staff were working remotely.
Chief executive Richard Lewis stressed no risks would be taken in putting on the tournament.
He said: "At the heart of our decision-making is our commitment to the health and safety of our members, staff, and the public, and we are grateful to the government and public health authorities for their advice and support.
"While we continue to plan for the championships at this time, it remains a continuously evolving situation and we will act responsibly, in the best interests of wider society.
"We thank all of our members, staff, players, partners, contractors and the public for their patience and trust as we continue to navigate this unprecedented global challenge."