Neither one is correct.
The correct forms are " Look forward to speaking with you" and "[I'm] Looking forward to speaking with you."
This is because you are looking forward to an event, and "speaking," as such, is a gerund, which is treated as a noun.
So you are actually "Looking forward to [the act of] speaking with you."
You could also rephrase it as "I hope to speak with you soon."
The correct forms are " Look forward to speaking with you" and "[I'm] Looking forward to speaking with you."
This is because you are looking forward to an event, and "speaking," as such, is a gerund, which is treated as a noun.
So you are actually "Looking forward to [the act of] speaking with you."
You could also rephrase it as "I hope to speak with you soon."
