Kappa with inverse-square weights gives you the same answer as the type 1 ICC, which I think is probably the best option simply because it's hard to articulate an interpretation for a given value of W (though I would love to hear people try!). 

Another option for ordinal data is the rank-correlation (Spearman).

I also suggest to make a graphical representation of the results and interpret the picture. Examples are heatmaps or scatterplots (see attachments).

Note: the symbol size in the scatterplot and the color intensity in the heatmap indicates the number of corresponding ratings.

Cohen's Kappa may suit?

Cohen’s Kappa (1960) is one of the most widely used indices of rater

agreement it overcomes the problem of chance agreement. It gives a value of between 0 and 1. 0 being the lowest level of agreement 1 being the highest. There are clear benchmarks for the strength of agreement have been constructed

for its use to aid in communicability. However it is susceptible to marginal homogeneity.

There is also Gwet's AC1 (2008) this is an alternative statistic. Like the Kappa it takes a value between 0 and 1. It is recommended that the same benchmark scales as those for the Kappa can be used for communication of levels of agreement. It is not sensitive to marginal homogeneity and positively biases for trait prevalence. It can be extended to multiple raters; It can deal with both nominal and ordinal data and it can deal with missing data.

I think what kind of data you are analyzing might matter here.  What kinds of items?  What construct?  What makes most sense might vary depending on whether the ratings are opinions that could reasonably vary or whether there is some known value that all raters should agree upon.  You might be setting too high a bar if you use a statistic like Kappa, depending on what is being measured.

The following paper might be helpful:

Shrout, P. E., & Fleiss, J. L. (1979). Intraclass correlations: Uses in assessing rater reliability. Psychological Bulletin, 86(2), 420-428. doi: http://dx.doi.org/10.1037/0033-2909.86.2.420

My readings suggest Cohen's Kappa provides levels of agreement between raters for nominal ratings. Also it may be limited in its usage with two raters only. What about cases with multiple raters and for Likert rating scale items which are considered as interval or ratio data? I am primarily interested in exploring the degree of consistency in ratings per item across the raters, I was suggested Kendall Coefficient of concordance. Is that appropriate? Please help ...

Your readings then don't include the calculation of weighted kappa. Using inverse-square weightings is commonly done, as this greatly increases the penalties for extreme disagreements, as well as giving a statistic that can be interpreted as an intraclass correlation

Thank you João Tiago Oliveira , that's one of the paper I have read in this respect.