"I like to play, I like to enjoy playing and I want to be the number one, but I have to deserve it as well," Adrian told West Ham's website http://www.whufc.com/news/articles/2018/may/21-may/adrian-i-want-be-west-ham-uniteds-no1-next-season.
Adrian, who was called up to Spain's senior squad in 2016 but has yet to make his international debut, kept six clean sheets in 19 league appearances this past season.
The 31-year-old ended the campaign with strong displays as West Ham collected seven points from their final three games to secure their top-flight status.
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Hart, 31, had a disappointing season on loan from champions Manchester City, conceding 39 goals while keeping four clean sheets in 19 appearances.
"This past season for me has been a bit disappointing and frustrating at times, but I have one more year on my contract and we'll see more when the season has finished," Adrian added.
"I hope that I can continue to show I am here to help West Ham, as I enjoy it here at London Stadium and I want to keep enjoying it."
West Ham also parted ways with manager David Moyes last week and are expected to announce former Manchester City coach Manuel Pellegrini as his replacement in the coming days.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by John O'Brien)
