"press strongly" is correct and usable in written English.
You can use it when you want to emphasize that something should be done with force and vigor. For example, "We need to press strongly to get the bill through Congress before the end of the year."
Developing countries will press strongly in Lima for solid pre-2020 comminments in accordance with scientific assessments.
It could also push other parties unhappy about the election's outcome to press strongly for an even broader recount.
In the lead-up to the 1988 summer games in Seoul, public protests against South Korea's military dictatorship escalated, prompting the I.O.C. to press strongly for a democratic transition.
Developing countries will press strongly for assurances that the contributions to the GCF are raised to at least $15bn – the minimum necessary, they say, for making it a credible institution.
In a letter to the Prime Minister, David Mepham, the group's UK director, and Hugh Williamson, its executive director, urged him to "take the opportunity to press strongly on a range of human rights concerns".
The mainstream press, strongly in favour of support for the Americans, complained that the government had now achieved Turkey's complete isolation.So is this the end of a steady friendship between Turkey and its most important ally?
If Van Gaal had stayed on, David de Gea would have pressed strongly to leave.
Other countries, pressed strongly by the powerful GM lobby, would probably follow, leading inevitably to more protests.
Under his guidance, the federation pressed strongly for strengthening the Clean Air Act, cutting toxic discharges into the Great Lakes and returning wolves to Yellowstone National Park.
There was an important group of top Nigerian civil servants and lawyers who pressed strongly for preserving unity, and whose influence was paramount.
In Clarke's telling, from the first hours after the attacks the President and other Administration officials pressed strongly for a connection to Iraq.
