Godfrey had surgery on Monday the right leg he fractured in the 1-0 defeat by Chelsea at the weekend. Everton said the 24-year-old was "expected to be out for around three months."
Another defender, Yerry Mina, suffered an ankle ligament injury in the second half of the game and will see a specialist this week to discover the full extent of the problem.
The injuries left Everton urgently seeking defensive reinforcements and Merseysider Coady, a Liverpool fan and former academy player, was happy to answer the call.
"I've grown up around the city," he said. "I know the football club, how big this club is and what it means to the supporters.
"I'm someone who was desperate to come here, to play for this club. I've got family and friends who are massive Evertonians."
Coady, who has played 196 times out of a possible 198 league games in the last five seasons, has captained Wolves for the last four years and has 10 England caps.
It is understood the Toffees have an option to buy at the end of the campaign.
Everton, who are also without injured striker Dominic Calvert-Lewin for up to six weeks, are set to announce deals for Lille midfielder Amadou Onana and former Blues midfielder Idrissa Gueye, who is currently at Paris St Germain,
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