'press the argument' is a correct and usable phrase in written English.
It is typically used in contexts where a speaker or writer encourages someone else to make a case for their point in a clear and concise manner. For example, "I can see that you have strong opinions on this issue, so please press your argument."
Hearing that negotiations for the author's latest novel were stalled, he jumped in to press the argument that Knopf had a successful history publishing Latin American writers.
Trump, meanwhile, has continued to press the argument that the media has given a misleading account of the crowds that attended his inauguration.
In particular, the United States did not press the argument that the Federal Government functionally holds title to the requisite interest because of the navigational servitude.
One is to press the argument, supported by many economists, that Germany benefits enormously from preserving the euro in its current form rather than abandoning it or standing by as it collapses.
— Larry No, and the opponents of same-sex marriage who press the argument based on procreation say it is the general ability of heterosexual couples to conceive children and not the particular abilities of a given couple that matter.
It is naturally eager to press the argument that reliance on American protection is no longer of any avail to small countries, that the time has come to denounce alliances and liquidate America's bases in Europe and Asia.
By John Cassidy March 14, 2016 With the next round of primaries looming, Bernie Sanders appeared Monday at a rally in Youngstown, Ohio, to press the arguments about trade that worked so well for him in Michigan.
Obama campaign officials made it clear that they would continue to press the arguments about Mr. Romney's record on women's issues in advertisements, and they spent an additional $7 million on a final-stretch television blitz that was already costing $40 million.
At both stops Thursday, Mr. Obama pressed the argument that Mr. Romney's plans would hurt older Americans.
Mr. Cunningham pressed the argument that as the South rapidly urbanized, Progressive Farmer's potential growth in readership was limited.
Mr. Greifeld again pressed the argument that the Securities and Exchange Commission would prevent a combined Nasdaq-NYSE Euronext from gouging customers on listings fees.
