Senior US Republicans have self-quarantined after coming into contact with a person who tested positive for coronavirus.
President Donald Trump, who also attended The Conservative Political Action Conference (CPAC) at the end of February, insists that he is in good health and has not taken a test.
While he did not interact with the infected person, Mr Trump shook hands with a convention host Matt Schlapp, who revealed he had previously had "incidental contact" with a patient.
Mr Cruz, Arizona Congressman Paul Gosar, Georgia Congressman Doug Collins and Florida Congressman Matt Gaetz are all self-quarantining for the recommended 14-days, after meeting the infected person at CPAC.
Mr Trump's newly appointed chief of staff, Mark Meadows, has also self-isolated after meeting the patient.
Meanwhile, the White House announced on Monday that Mr Trump had not been tested for the virus.
"He has neither had prolonged close contact with any known confirmed Covid-19 patients, nor does he have any symptoms," a spokeswoman said.
"President Trump remains in excellent health, and his physician will continue to closely monitor him."
Coronavirus - In pictures
There are more than 700 confirmed cases in the US and 26 people have died so far.
The White House also warned on Monday that the number of cases in the US was likely to rise.
"There will be more cases," Mr Pence told reporters. "But we simply ask today for the American people to engage in the common sense practices."
The vice-president, who also confirmed he had not been tested for the virus, said the administration was consulting Congress on providing paid sick leave to workers.
