Alonso has revealed his frustration with the lack of progress made by the team over the past couple of years and there were reports suggesting he was being targeted by Ferrari and Mercedes.
However, McLaren recently confirmed they will be splitting with engine supplier Honda at the end of the current season and joining forces with Renault instead, ending a disappointing three-year partnership.
With the change in power-unit supplier, Brown now believes Alonso will sign a new contract with the team and stay for at least 2018 as the team looks to close the gap.
"We're very confident we'll have Fernando Alonso in our car next season and the same driver line-up that we have now," he told Sky Sports F1. "Fernando is a very important asset, he's very happy with the decision, we're happy with him in the car, he's happy being in the car and I think we'll have a good ending to this story."