The phrase "asserts that they will" is correct and usable in written English.
It can be used when discussing someone's claims or statements about future actions or intentions.
Example: "The report asserts that they will implement the new policy by the end of the year."
Alternatives: "claims that they will" or "states that they will".
A recent article in Forbes on peer-to-peer rentals asserts that they will grow 25 percent this year, "moving from an income boost in a stagnant wage market into a disruptive economic force".
Working-class voters are the ones he believes President Bush has abandoned, and he asserts that they will trust John Edwards, just as North Carolina juries have trusted John Edwards in a dozen big cases.
He asserts that they will never amount to anything if they don't put down their "chicken wings and malt liquor" and if they continue to support a "former running back who stabbed his wife".
They assert that they will have cap-and-trade, with a cap that reduces emissions by whatever amount is needed.
Hezbollah leaders have asserted that they will continue fighting Israel until it releases the two guerrillas and other Lebanese prisoners.
American and British senior officials still assert that they will prevail and seem intent on sticking with their plan.
But Mr. Bradley's aides assert that they will be able to place experienced field operatives in every important state.
On both issues, in fact, many House Republicans are proudly asserting that they will stand in the way of any attempts to conduct a conference with the Senate.
Leaders in both parties have repeatedly asserted that they will reach agreement and avoid a government shutdown, although lawmakers have missed all their other deadlines so far.
Is there a contradiction in telling schools they will have more freedom, but also asserting that they will be "properly accountable"?
