Detroit — One name invoked frequently during Wednesday night's debate was that of former President Barack Obama, who has no plans to participate in the Democratic presidential campaign this year but is keeping tabs on the contest.

A source familiar with his thinking wouldn't say whether the former president watched Tuesday and Wednesday's debates, but said Obama believes "his legacy is going to be fine, that there's a staying power to it and the things under attack by this president are high water marks for the country."

The person familiar with the former president's thinking was granted anonymity to speak frankly about the politically sensitive subject.

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Obama remains extremely popular among Democrats, most of whom remember his presidency very fondly. In a CBS News poll in May, 58% of Democrats nationwide rated his tenure as "excellent," with another 33% saying it was "good." Just 9% rated his time in office as "fair" or "poor."

President Obama's presidency was highly rated by Democrats in a CBS News poll in May 2019. CBS News

The former president believes it's appropriate to review and debate his legacy, but believes it should be "fact-based," especially in regards to his deportation policy, the person said.

"The debate over immigration doesn't always seem to be rooted in reality," the person said.

During the eight years of the Obama administration, the Department of Homeland Security prioritized deporting criminals and people who had recently illegally crossed the border. The Trump administration has targeted family units and in several cases, deported people who have lived in the country for several years.

"It is perilous to the party to attack one of the most popular figures among voters in the country," the Obama associate added. "This primary has got to be about making the strongest possible case against Donald Trump and not be a circular firing squad. It could ultimately backfire on candidates who choose to engage in this rhetoric."

During Wednesday's debate, former Vice President Joe Biden frequently invoked his partnership with Obama, but repeatedly demurred when pressed by New York City Mayor Bill de Blasio and others on whether he'd ever raised concerns about the administration's deportation policy.

Biden's refusal to engage on the subject earned a powerful rebuke from Democratic Sen. Cory Booker of New Jersey, who told Biden, "You can't have it both ways. You invoke President Obama more than anybody in this campaign. You can't do it when it's convenient and then dodge it when it's not."

Campaigning Thursday in Detroit, Biden echoed the former president's concerns — although there was no sign of coordination between the two.

"I was surprised about how much incoming there was about Barack," Biden told reporters outside a Detroit hot dog restaurant where he met with the city's mayor and other local leaders.

Rebecca Kaplan and Stephanie Ramirez contributed to this report.