Currently those who behave in such a way face a breach of the peace charge.
Justice Secretary Kenny MacAskill wants a new offence outlawing "threatening, alarming or distressing behaviour".
The government will seek to change this by lodging an amendment to the Criminal Justice and Licensing (Scotland) Bill, which is going through parliament.
Once the new powers are enshrined in law they will give prosecutors the ability to act against stalkers who send threatening messages via email, text, phone calls and on internet sites such as Facebook.
'No escape'
Mr MacAskill said: "Stalking can be a deeply frightening crime for victims and we want to ensure that the small minority of perpetrators who engage in this criminal activity are brought to justice.
"We want to send out the message loud and clear that if you carry out this offence, there will be no escape, there will be no wriggle room to exploit and you will be met with by the full force of law."
A government spokesman said the proposed offence would cover not only the sending of threatening or harassing emails, text messages or phone calls, but also persistent following, pursuing or spying on someone.
He added that the new offence would also ensure that prosecutors can take action in other areas, such as incidents of domestic abuse that take place in isolated locations.
BBC News
