Which is correct – ‘Looking forward to speak with you’ or ‘Look forward to speak with you’?

Neither. “‘Look forward to speak with you'” or “Looking forward to speak with you.” The “to” in the sentences is a true preposition, not so much a pro forma “to” that precedes an infinitive.

It’s worth noting that “looking forward” is a phrasal verb based on these two common English category-forming notions. Looking forward, symbolically, means to pay attention to or be aware of something, and looking forward is where the future is. 

Its literal meaning is “anticipate,” but it has acquired a positive connotation in language.

Neither is official English, and both are missing terms that I’ve included in brackets.

I am looking forward to speaking with you.

I look forward to speak with you.

The subject-assisting verb I am is inferred in the first construction. I’m looking forward to it… because it sounds more familiar and relaxed than the other.

Examples

• I’m looking forward to speaking with you this weekend.
• I’m looking forward to meeting you at the party and learning more about you.

The subject I is the only one who is implied in the second. I’m looking forward to… because it sounds more formal and professional than the other.

Examples

I hope to get the opportunity to speak with you soon. Alternatively, I hope to hear from you soon. Sounds like the final sentence of a job application cover letter.

That isn’t to say that one is always and only formal and the other is always and only informal, and that one sounds more formal than the other in comparison.

A school bully, for example, could use either to encourage you to meet him after school, depending on his posture, expression, & tone of voice, among other factors:

School bully 1

 Punk, I’m looking forward to thrashing you after school. He has a scary, forceful, and authoritative tone to him. I’m standing still, and his face is expressionless.

School bully 2

Punk, I’m looking forward to thrashing you after school! He’s having a good time, with a nasty grin on his face. He’s rubbing his hands together, ready to bash me up with them.

These are just instances of how I hear the difference, not absolutes!

In English (unlike, say, Spanish), omitting these words is uncommon, but it can be done in pleasant contexts when the intention is obvious. “Look forward to” is a little more active than “look forward to,” but both are fine.