Soon after Rajeev Suri was named chief executive of Nokia Corp. last year, he sat down with Alcatel-Lucent CEO Michel Combes to discuss the possible purchase of his French rival’s wireless business, according to people familiar with the matter. But Mr. Combes preferred another idea: selling the entire company.

On Wednesday, Mr. Combes’s point of view won the day, when Nokia agreed to buy out Alcatel-Lucent in an all-stock deal that values the French firm at €15.6 billion ($16.6 billion).

The deal caps a pursuit that stretches back to 2013, based on the logic that Europe’s two smaller network-gear makers would eventually need to team up in order to have the resources to compete globally against larger rivals including Sweden’s Ericsson and China’s Huawei Technologies Co.

In the end, both sides decided splitting off Alcatel’s business making wireless networks would be too complicated, and would deny both companies the advantages of selling everything from Internet backbone equipment to cellular antennae, some of the people familiar with the matter said.

“To compete in the future you will need end-to-end solutions across the whole portfolio,” Mr. Combes said Wednesday. “My strong preference was to combine the two companies.”