NYT notes the expanded role internally of Kellyanne Conway in the selection process.She is a newly appointed senior adviser in the Trump campaign. She is Founder and President of the polling company, inc./WomanTrend and appears close to Newt Gingrich and has also done work for Ted Cruz.
At this point, all indications are that Trump's short list is composed of serious war hawks.
The latest named to be rumored to be on Trump's short list in addition to war hawks Christie, Gingrich and Sessions is Indiana Gov. Mike Spence (Trump is meeting with Spence this weekend).
According to Wikipedia:
Pence supported the joint resolution authorizing military action against Iraq, otherwise known as the Iraq Resolution...
During the Iraq War, he opposed setting a public withdrawal date from Iraq. During an April 2007 visit to Baghdad, Pence and John McCain visited Shorja market, the site of a deadly attack in February, 2007, that claimed the lives of 61 people. During the visit, Pence and Senator John McCain were accompanied by General David Petraeus. Pence and McCain described the visit as evidence that the security situation in Iraqi markets have improved. The visit was criticized by the New York Times as giving a false indication of how secure the area was due to the extremely heavy security forces McCain brought with him. The visit to the market took place under large security including helicopters overhead.
Pence has opposed closing the Guantanamo Bay detention camp and trying the suspected terrorists in the U.S. Pence believes that “the Obama administration must overturn this wrongheaded decision”. As an alternative, Pence has said that the "enemy combatants" should be tried in a military tribunal.
He has stated his support of Israel has defended the actions of Israel in its use of deadly force in enforcing the blockade of Gaza and has referred to Israel as "America's most cherished ally".