Kobe Bryant to Vanessa Bryant: I Won't Cheat on You Ever Again, I Swear!!

Kobe Bryant is going all out to save his marriage.

The L.A. Lakers star has reportedly sworn up and down to wife Vanessa Bryant that he will never, ever stray again if she calls off their divorce and takes him back.

Kobe has "aggressively pursued" Vanessa over the last 2-3 weeks, and she's been pretty receptive ... but she still isn't sure the marriage can be resurrected. 

When we say aggressively, we don't just mean promises of faithfulness. We mean gifts. He's reportedly given Vanessa a VERY expensive piece of jewelry.

Kobe famously gave Vanessa a $4 million "adultery ring" after she stood by him during his infamous affair / rape trial in 2003 (Bryant was acquitted of all charges).

As for whether they will drop the divorce - the terms of which are finalized but which does not become official for months - they are working on a reconciliation.

Sources very familiar with the couple say they both love each other and not to rule out the possibility that the marriage will be saved, despite Kobe's cheating.

On the flip side, Vanessa Bryant and the couple's two kids were in New York City over the weekend, looking for a big apartment ... not exactly a good sign.

Vanessa, now worth at least $75 million, won't say if she's packing up and moving from California for good, but she definitely wants to buy a place in NYC.

Kobe reportedly knows she's shopping in the Big Apple, and isn't happy about it. Could she be planning on skipping town for good? Using this as leverage?

[Photo: WENN.com]