Linked Server stops responding for particular login We have an issue where the Linked Server does not work for the app team login but tests successfully when the DBA checks the connectivity. On investigating we found that the app login had some Killed/Rollback sessions which are causing this issue. The only resolution we have now is to Restart the instance.

Multiple sessions from the app login which are in Killed/Rollback and would not do anything (0 CPU Time and 0 Disk IO). This is a report login which extracts reports all day and sometimes this happens.

Is there any way to fix it without an instance restart?
Stop restarting SQL Server. Just stop doing that. If you have a particular problem with a particular process that is being killed and rolling back, let it finish. Then, find out what's causing that to happen. Use Extended Events, or a server-side trace, heck, even a third party tool like sp_WhoIsActive. Do something.

SQL Server has a transaction log. All modifications go in there. When these are committed, the changes go into the data file. When they are killed, they have to roll back. 

That's just how it works. The rollback has to finish for the database to remain transactionally consistent.

Restarting SQL Server does not make this go faster. In fact, you could easily make things worse, possibly even taking the database offline or corrupting it.

Admittedly, this answer does not address your question, but there's not enough space in the comments to explain why you need to figure out what's causing the killed processes. It could be a DBA, it could be an automated process.

Bottom line: let SQL Server finish what it's doing. If it takes a long time to do something, killing the task will take just as long, if not longer, as that to roll back.