Activist investor Ryan Cohen has built a stake in Alibaba Group Holding Ltd. value a whole lot of thousands and thousands of dollars and is privately pushing the Chinese e-commerce giant to speed up and further boost its share-repurchase program, in response to people acquainted with the matter. 

Mr. Cohen, often called the meme-stock king for helping ignite explosive rallies in GameStop Corp. and others, built the stake within the second half of last yr, the people said.