The states, selected from 16 finalists announced earlier this month, received hundreds of millions in grants designed to encourage the use of innovative programs to improve student performance and transform struggling schools.
It wasn't immediately known exactly how much money the two states would receive, but officials in Tennessee said they applied for $500 million and their counterparts in Delaware asked for $100 million.
The source declined to be identified because not all finalists had been yet contacted.
The winners beat out: Colorado, the District of Columbia, Florida, Georgia, Illinois, Kentucky, Louisiana, Massachusetts, New York, North Carolina, Ohio, Pennsylvania, Rhode Island and South Carolina.
U.S. Department of Education officials are expected to hold a press conference later Monday to talk about the winners. Education Secretary Arne Duncan was making calls to each finalist Monday morning to let them know the results.
The money is part of President Barack Obama's economic stimulus law, which provided a staggering $100 billion for schools. The $4.35 billion is part of that larger allocation.
The Education Department asked states to concentrate their proposals on four areas: adopting standards and assessments to better prepare students for careers and college; getting high-quality teachers into classroom; turning around low-performing schools; and creating data systems to track performance.
Federal officials will collect a second round of applications for the highly selective grant program in June. The states that were not picked this time can reapply for grants then.
More than 40 states applied for the grants, scrambling to widen charter school laws and enact performance pay for teachers to prove that they deserved part of the money.
Some education observers have criticized the "Race to the Top" competition, saying the administration is out of touch because it is pushing reform at a time when states can barely afford basic necessities and are laying off teachers by the hundreds.