The baby's umbilical cord is attached to the placenta, not your belly button. He/she is likely just moving and causing an odd sensation when he rolls against your belly button. It's a strange feeling!
I know what pain you're talking about and no, that's not from the baby tugging on the umbilical chord. I personally think it's from your skin stretching. Try applying something cold on it, that seems to help me.
You definitely can't feel the baby pulling on his/her umbilical cord because it isn't directly attached to you. But, you may have a stretched belly button, which is very tender when babies roll around in there. I have quite a bit of discomfort and some occasional pain surrounding my navel, particularly when the baby moves and pushes really hard against my skin, and I didn't have that pain during my first full-term pregnancy. It's totally normal.
Natural miscarriage @ 8 weeks - 3/8/2005
Big Brother "Skippy" born - 2/28/2007
Missed miscarriage - (EDD 3/5/2013) - D&E @ 11 weeks - 8/8/2012
"Hen" (EDD 6/7/2013) - born sleeping @ 19 weeks - 1/15/2013
"G-Unit" born - 4/14/2014 and he's 100% perfection!!
Re: Is it possible to feel the baby tugging on the umbilical chord?
Natural miscarriage @ 8 weeks - 3/8/2005
Big Brother "Skippy" born - 2/28/2007
Missed miscarriage - (EDD 3/5/2013) - D&E @ 11 weeks - 8/8/2012
"Hen" (EDD 6/7/2013) - born sleeping @ 19 weeks - 1/15/2013
"G-Unit" born - 4/14/2014 and he's 100% perfection!!