Following City's 2-0 Premier League win at Vitality Stadium, Guardiola embraced Arter on the pitch - and the 27-year-old called his words "a lovely touch".
"It's great for him to take his time out to speak to me, I really appreciate it," Arter told BBC Radio Solent.
He and partner Rachel's baby daughter, Renee, was stillborn in December 2015.
In October, on Football Focus, it was announced the couple were expecting another child, with Rachel 22 weeks pregnant.
Speaking about Spaniard Guardiola, Arter added: "He offered me and my partner his best wishes for the weeks ahead.
"He's someone I have massive respect for. I watched his Barcelona teams and he is, without doubt, the best manager in the world."
Guardiola was asked at his post-match news conference about what he had said to Arter and replied: "He will be a father soon so I said big congratulations to him and his partner.
"I know what happened in the past, and, hopefully, the baby will be well."
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Bournemouth's official Twitter account praised Guardiola for a "touch of class" in seeking out Arter after Monday's game.
City moved up to second in the table after the hard-fought victory, earned through a Raheem Sterling strike and a Tyrone Mings own goal.
The defeat leaves Bournemouth in 14th place, six points above the relegation zone.
Bournemouth midfielder Harry Arter was touched to receive the wishes of Manchester City manager Pep Guardiola as his partner prepares to give birth.