According to several news media outlets, the answer is no, the Pope can not have pets. It has been reported that the policies for residing in the Vatican apartments state that pets are not allowed.
However, there are a few Popes who have had pets while leading the Church. Pope Leo XII kept a small dog for company, several Popes had aviaries built in the Vatican gardens, and Pope Pius XII kept caged birds in the papal apartment. You also have Pope Leo X, who was gifted a white elephant, named Hanno, by King Manuel I of Portugal.
Pope Emeritus Benedict XVI is well-known for his fondness of cats. When asked about his affection for cats, Cardinal Mahony commented,
The street talk that the pope loves cats is incorrect. The pope adores cats.
Many have wondered hopefully if Pope Emeritus Benedict XVI will be able to have a pet cat now that he has resigned from the papacy. The answer remains unclear, but it has been reported that outside the newly-renovated monastery, where he will reside, there are already many cats around that he can look after.