Pup Trevor disappeared after his owner Claire Rodgers let him out into the garden in Gipton on Thursday morning.
But when she returned to let him back in, the two-year-old pug had vanished.
It's thought Trevor may have escaped, although Claire and husband Scott say they now fear he may have been picked up due to his pedigree.
Two days from her due date on Saturday, Claire was forced to wander the estate shouting for him, and says she is "so worried".
"We put him outside yesterday and I came back out and he wasn't there. He has either escaped or been taken, or a combination of the two.
"I just panicked when I noticed he'd gone. I was so worried, especially as I'm due to give birth any day."
She added: "We have done everything. I am worried because he is a little stupid, friendly dog and if he sees anyone he will happily go up to them and want to be their friend. He is our little fur baby."
Anyone who spots Trevor or knows any information on his whereabouts should contact the Searching for Trevor Pug Facebook group.