McConnell gave a half-hour speech on Dec 19, in which he expressed his opposition to allowing the Democrats to call anymore witnesses, quite a few times. In the speech, he said (numerous times) that the Democrats have rushed the impeachment process (in order to fit their political timetable/agenda) by not pursuing all the necessary witnesses and information through court battles, drawing an analogy with Nixon's impeachment, which McConnell says took over a year to prepare, including the (well known) supreme court case US vs Nixon. (In contrast, he argued that the Democrats only spent 12 weeks preparing Trump's [current] impeachment articles.) McConnell also argued in his speech that allowing the Democrats to present more witnesses in the Senate trial would be basically doing their [impeachment preparation] job for them... and also opening the floodgates for future half-baked impeachment attempts, which he argues the present Senate has a historical duty to thus prevent. In relation to this, he also dismissed the Democrats' second article of impeachment (obstruction of Congress) by saying that disagreements over executive privilege happen all the time, in all modern presidencies. (He also dismissed the first article, but that's besides the point here.)
I haven't seen any clear indication in that speech on whether he will allow Trump's witnesses, but McConnell has also reiterated what is (by now) the standard Trump defense line that the process (in the House) has been unfair to the president.
Schumer gave an equally long speech on the same day in which he rebutts some of McConnell's points. AP chose to highlight this phrase repeated on both sides of the exchange:
On the Senate floor, McConnell described the House actions against Trump as “the most rushed, least thorough and most unfair impeachment inquiry in modern history.”
Fighting back using McConnell’s own words, Schumer said the Republican leader was plotting the “most rushed, least thorough and most unfair” impeachment trial in history by declining to agree to call witnesses, including former Trump national security adviser John Bolton, who declined to testify before the House.
A Politico article notes that the two men met face-to-face later in the day, but failed to resolve their differences on how the impeachment trial should be structured. The same piece notes that Pelosi said the House won't be formally sending the articles/managers to the Senate because they “haven’t seen anything that looks fair to us” [presumably in terms of the trial rules.]