In the picture, West can be seen checking his phone as he holds his daughter close.
Kardashian West congratulated the rapper on his return to music following the surprise release of his eighth studio album on June 1.
Alongside the picture, Kardashian wrote on Instagram: “Happy Birthday babe! It’s been a wild year but we’re here and life is good!
“So proud of you dropping back to back albums and fulfilling your dreams.
“You inspire me to make the impossible always become a reality. Love you.”
The picture had attracted more than 500,000 likes on Instagram within half an hour.
Earlier this week, Kardashian West said she would be holding a small dinner party to celebrate West’s birthday.
Speaking after receiving an Influencer Award at the CFDA Fashion Awards on Monday, Kardashian West told US celebrity show E! News: “I’m planning a dinner for him at our house.
Chicago, the couple’s third child, was born in January of this year. They also have an older daughter, four-year-old North, and a son, two-year-old Saint.