Attempts by state legislators
Yes, it has been proposed by various legislators at the state level.
According to Houston Public Media:
A bill (HB 1938) recently filed by State Representative Jason Villalba of Dallas would automatically make Texans organ donors — unless they opt out. The bill would apply to first-time driver's license applicants and renewals who are 18 and over. Unless they specifically opt out on the form, citizens would automatically be added to organ donor registry. If passed, the bill would be the first of its kind in the United States.
I assume the bill didn't pass, because an article on thehumanist.com in 2018 read:
Still, efforts at organ donation consent policy reform in the US have been fruitless. Colorado, Texas and Pennsylvania legislators, for example, have pushed for presumed consent. Ted Kennedy Jr.’s attempt at introducing an organ donation “opt-out” policy in Connecticut was met with strong backlash just last year. The engrained American value of individual rights superseding that of a menacing government could be the source of this initial pushback, but one would like to believe it could be overcome if the American public was more aware of the current organ shortage gap.
In addition to the states named there, a page on the US Department of Health & Human Services' website also mentions the state of Maryland as having tried to pass such a presumed consent law:
Policies based on presumed consent are seen in many states' statutes bearing on the retrieval of corneas. Regarding organ and tissue donation, two presumed consent bills were submitted in early 1993 to the state legislatures in Maryland and Pennsylvania, the Maryland version having been introduced and rejected in a previous session.
Nationwide efforts by civilians
Online, I found a few petitions as well. Most petitions were aimed at other countries, but I found two aimed at US legislators as well.
This petition aimed at the US Congress received 153 voters. It called on legislators to:
Change the current organ donation system to a presumed consent or “opt-out” program instead of the current “opt-in” program in order to create a larger number of available organs and save lives while still allowing people who wish to not become donors to “opt-out” of donating simply by visiting a website or registering as a non-donor with their local DMV or hospital.
Another petition in 2013 received 1757 signatures on the White House's petition website. It called for legislators to:
Pass a Bill to Change the United States' Organ Donation System to be "Opt-Out" Rather Than "Opt-In"