(CNN) Senate Republicans aren't ready to embrace the White House's proposal to overhaul the nation's legal immigration system, amid skepticism about elements of the plan and the timing of it.

At a closed-door lunch Wednesday, Trump's son-in-law and senior adviser Jared Kushner, who spearheaded the proposal, faced questions from GOP senators about the range of immigration matters that are left out of his plan, with some arguing that the plan needs to address other matters in order to get broader support.

Kushner's argument: Focus on issues that could get broader support. But Republicans argue that it's simply not feasible since diving into immigration opens up a range of hot-button issues that have long divided Congress, namely what to do about illegal immigration, and those are inevitably going to reemerge.

"We all know what those are," said Republican Sen. Mike Braun of Indiana, who supports increasing legal immigration. "He said they aren't going to disregard those. He said we are going to focus on this because this ought to be something we all can agree to. Why would the Democrats not be for more legal immigration with the focus on who we are bringing in?"

Others made clear that they need to review the details.

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