Britney Spears is feeling nostalgic about family, including her dad Jamie Spears, and it's the latest sign she's warming to the idea of a father/daughter reconciliation.

The singer posted a slew of throwback family snaps Tuesday -- most showing her carrying her now-teenage sons when they were much younger, but one also shows Jamie next to Brit ... holding his grandson's lunchbox and a juice box.

Britney didn't caption the photos, but as we told you, she's opening the door to the possibility of reaching out to her father -- so, the pic could be seen as her extending an olive branch to Jamie.

TMZ broke the story ... doctors had to amputate one of Jamie's legs after 5 unsuccessful surgeries to combat a massive infection. In addition to his ailing health, Jamie's had to cope with the barrage of criticism over his handling of Britney's conservatorship -- with the singer publicly blasting him on social media herself.

However, we were told the star softened towards him recently ... even missing him sometimes and considered sending him money to help him out.

We're told Jamie's totally open to reconnecting with his daughter, and would love nothing more than to be on good terms again with Brit.

As you know, she's also been getting along with mom Lynne, who flew out for Britney's 42nd birthday bash over the weekend and she's also communicating with sister Jamie Lynn.