The 23-year-old was in impressive form again despite the relegated Royals going down 2-0 to Manchester City on Tuesday, and is expected to attract interest from several clubs over the summer.
But Adkins said: "I'm sure there will lots of speculation. But we are in a good position in respect of the finances and there is good stability here. We're not looking to lose Alex and are looking to build for the future.
"He made some excellent saves and has demonstrated that on a few times now. We want to build and play from the back and one or two of his distribution choices weren't the greatest, but others were.
"We believe he can be a top-four keeper at international level and, if you can play out from the back, he can do that.
"We want to build and play from the back and play through the field of play. From my point of view, everything is very positive moving forward and we have a lot of players who have won the Championship before."