It would be possible to design a whistle with recorder fingerings that would work at least for the first two octaves, maybe even for the third. It would be necessary to lengthen the tube and add an extra hole for the RH 4th finger, and add a hole for the LH thumb. Obviously there is no demand for such an instrument because as far as I know no instrument maker builds one. Of course it's possible that recorder players might have had whistles custom made for them.  

The opposite, a [recorder with whistle fingerings](https://music.stackexchange.com/a/99777/40545), is commercially available.
 
Another answer here states that a whistle with recorder fingerings would be a recorder. This is not the case. There are important differences between a whistle and a recorder. The main factors affecting the differences in sound are:
 - Bore design.   
Whistles are normally cylindrical, recorders are reverse conical. A typical alto recorder has a bore of about 17mm at the top of the main joint to 10mm at the foot. There is more variation in whistle bores, but 17mm is about average for a low G whistle (which would correspond to an alto recorder). This gives the recorder a much 'sweeter' sound, and it's the reason a whistle will never sound the same as a recorder. There are some whistles (Shaw) with a conical bore, but they have a very different fipple design and sound very 'breathy'.    
 - Fipple design.  
There is a lot of variation in the fipples of whistles. Some are quite close to the recorder design, and these whistles tend get closer to the recorder sound.
 - Tube material.  
Recorders are made of wood (or plastic). Whistles are usually made of various metals. Whistles made of wood or plastic also get closer the the recorder sound.
 - Finger hole size  
Whistles generally have larger finger holes than recorders which gives them a louder, brighter sound. The disadvantage is that it's harder or impossible to cross-finger for chromatic notes. Some whistles do have smaller holes which make them a sound slightly more recorder-like, but allows for good cross-fingerings.