No you don not the ends are not compatible. You need to increase the digestion time since BsiWI has a higher activity at 50C but will still work at 37C just at a slower rate. I would suggest overnight digestion to be sure, followed by gel purification.

There are two problems with using those two specific enzymes in a single digestion reaction :

1. Optimal incubation time differs (37C for NheI, 55C for BsiWI

2. Incompatible optimal buffer activity (maximum activity of 50% for BsIWI in NEBuffer2.1).

If you get self ligation, it's because most of your DNA is only digested by NheI, making it possible for recircularization. I strongly recommend using sequential digestion and not dephosphorylating the vetor. Follow this detailed protocol :

http://nebcloner.neb.com/#!/protocol/re/sequential-purify/NheI,BsiWI

Finally, please note that some people recommend a gel purification step prior to ligation with your insert to get rid of the small fragment you cut out (which can bind to your insert and lower your chances of successfull ligation). I personally recommend running the digestion product on an agarose gel to make sure only one band is present (if you have several bands, it could mean you still have some supercoiled/non-digested plasmid left). To minimize loss of sample, you can do a simple PCR purification step.

Thank you both. I will take your advise into consideration.