The best explanation of the president’s role in Supreme Court justice nominations is: "The president makes the final nominations to the Senate after they decide on the best candidates."
However, it's important to clarify that, in reality, the president has the authority to nominate justices to the Supreme Court, and then the Senate has the responsibility to confirm or reject those nominations. The statement you've referenced suggests a collaborative decision-making process involving the Senate before presidential nominations, which is not accurate; the president acts unilaterally in making the nominations.