Premier League clubs reached an agreement on Monday for players to return to training on Tuesday in small groups.
It was later revealed that the league had six positive coronavirus cases with Watford confirming three of them, including one unidentifed player.
"We're due back in this week. I've said I'm not going in," Deeney told boxing promoter Eddie Hearn and former WBC cruiserweight champion Tony Bellew.
"My son is five months and he's had breathing difficulties. I don't want to come home to put him in more danger. You've got to drive in in your own kit, you can't have showers, then you've got to drive back home in the same dirty kit you've got.
"While we are getting tested and while we are going to be in a very safe environment, it only takes one person to get infected within the group. I don't want to be bringing that home."
Watford were due to return to training on Tuesday alongside other clubs but manager Nigel Pearson told the Times on Sunday that he would not force any player to return. Pearson also expressed his concern at the effect of football coming back too early.
Deeney said he had also raised concerns about the increased risk to black, Asian and minority ethnic (BAME) players while in a meeting with the Premier League, other team captains and medical experts.
Britain's statistics office said this month that black men are 4.2 times more likely to die from coronavirus-related causes than white males.
About a third of Premier League players are from a BAME background.
"My problem was in the meeting, I asked very simple questions," he said
"For black, Asian and mixed ethnicities, they're four times more likely to get the illness, they're twice as likely to have long lasting illnesses -- is there anything extra, additional screening, heart stuff to see if people have got problems with that? No. OK, well I feel that should be addressed.
"I can't get a haircut until mid-July but I can go and get in a box with 19 people and go and jump for a header and nobody could answer the questions, not because they didn't want to, just because they don't know the information."
Despite agreeing a date to return to training, the Premier League has yet to set a return date. Professional football has been suspended in England since March 13.
Clubs will meet again next Tuesday to discuss a return date. "Project Restart" has hit several stumbling blocks including concern for player welfare and disagreement on whether games should be played at neutral grounds or not.
Information from Reuters contributed to this report.